COMPRESSED  AIR 


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COMPRESSED  AIR 


THEORY  AND  COMPUTATIONS 


BY 


ELMO  G.  HARRIS,  C.E. 

PROFESSOR   OP   CIVIL  ENGINEERING,    MISSOURI   SCHOOL   OF   MINES, 
IN   CHARGE   OP   COMPRESSED   AIR   AND   HYDRAULICS; 
MEMBER   OF  AMERICAN   SOCIETY    OF 
CIVIL   ENGINEERS 


SECOND  EDITION 
REVISED  AND  ENLARGED 


McGRAW-HILL  BOOK  COMPANY,  INC 

239  WEST  39TH  STREET.    NEW  YORK 


LONDON:  HILL  PUBLISHING  CO.,  LTD. 
6  &  8  BOUVERIE  ST.,  E.  C. 

1917 


COPYRIGHT,  1910,  AND  1917,  BY  THE 
MCGRAW-HILL  BOOK  COMPANY,  INC. 


THE  MAPLE  PRESS  YORK  PA 


PREFACE  TO  SECOND  EDITION 

AFTER  five  years  trying-out  of  the  first  edition  the  second  has 
been  prepared  with  the  view  to  eliminate  all  errors  and  ambigui- 
ties and  to  add  matter  of  value  where  possible  without  burdening 
the  text  with  illustrations  and  descriptions  of  matter  of  only  tem- 
porary value,  such  as  machines  and  devices  that  are  in  use  today 
but  may  be  succeeded  by  better  ones  in  a  few  years.  It  is  the 
author's  opinion  that  current  practice,  in  the  general  form  of 
machines  and  their  details,  can  best  be  studied  in  trade  circulars, 
of  which  there  are  many  very  creditable  productions  illustrating 
and  describing  a  greater  variety  of  machines  than  can  possibly 
be  shown  in  a  text-book. 

A  new  chapter  has  been  added  on  centrifugal  fans  and  turbine 
compressors.  The  author  has  found  a  need  for  a  clear,  concise 
presentation  of  the  theory  underlying  such  machines,  and  be- 
lieves that  a  more  general  knowledge  of  the  technicalities  of  the 
subject  will  lead  to  material  betterment  of  the  cheaper  forms  of 
fans  that  make  up  the  greater  portion  of  the  total  in  use. 

Appendix  B,  on  design  of  Logarithmic  charts,  should  be  welcome 
to  most  students  since  such  matter  has  not  appeared  in  text- 
books in  common  use. 

Compressed  air  has  long  held  the  field  for  rock  drilling  under- 
ground, though  electricity  has  several  times  attempted  to  get  into 
that  business.  At  one  time  it  seemed  that  compressed  air  would 
prove  the  best  motive  power  for  underground  pumps,  but  in  more 
recent  years  the  improvements  in  centrifugal  pumps  seem  to  give 
electricity  the  advantage. 

In  general,  it  may  be  assumed  that  where  rotation  is  desired 
electricity  will  have  the  advantage,  while  where  rapid  reciprocat- 
ing motion  is  desired  air  will  have  the  advantage.  In  the  latter 
class  are  all  kinds  of  pneumatic  hammers,  which  have  revolution- 
ized several  industries  since  they  have  been  introduced. 

It  is  not  the  intention  to  enumerate  here  the  applications  of 
compressed  air.  It  is  a  very  versatile,  willing  and  good-natured 
servant.  It  offers  a  fascinating  field  for  the  inventor  and  its 
usefulness  and  already  numerous  applications  will  surely  increase. 

HOLLA,  Mo.,  E.  G.  HARRIS. 

April,  1917. 


367514 


PREFACE  TO  FIRST  EDITION 

THIS  volume  is  designed  to  present  the  mathematical  treat- 
ment of  the  problems  in  the  production  and  application  of  com- 
pressed air. 

It  is  the  author's  opinion  that  prerequisite  to  a  successful 
study  of  compressed  air  is  a  thorough  training  in  mathematics, 
including  calculus,  and  the  mathematical  sciences,  such  as 
physics,  mechanics,  hydraulics  and  thermodynamics. 

Therefore  no  attempt  has  been  made  to  adapt  this  volume  to 
the  use  of  the  self-made  mechanic  except  in  the  insertion  of  more 
complete  tables  than  usually  accompany  such  work.  Many 
phases  of  the  subject  are  elusive  and  difficult  to  see  clearly  even 
by  the  thoroughly  trained;  and  serious  blunders  are  liable  to  occur 
when  an  installation  is  designed  by  one  not  well  versed  in  the 
technicalities  of  the  subject. 

As  one  advocating  the  increased  application  of  compressed  air 
and  the  more  efficient  use  where  at  present  applied,  the  author  has 
prepared  this  volume  for  college-bred  men,  believing  that  such 
only,  and  only  the  best  of  such,  should  be  entrusted  with  the 
designing  of  compressed-air  installations. 

The  author  claims  originality  in  the  matter  in,  and  the  use  of, 
Tables  I,  II,  III,  V,  VI,  VII  and  IX,  in  the  chapter  on  friction 
in  air  pipes  and  in  the  chapter  on  the  air-lift  pump. 

Special  effort  has  been  made  to  give  examples  of  a  practical 
nature  illustrating  some  important  points  in  the  use  of  air  or 
bringing  out  some  principles  or  facts  not  usually  appreciated. 

Acknowledgment  is  herewith  made  to  Mr.  E.  P.  Seaver  for 
tables  of  Common  Logarithms  of  Numbers  taken  from  his 
Handbook. 


Vll 


CONTENTS 

PAGE 

PREFACE      v 

SYMBOLS xi 

FORMULAS xiii 

INTRODUCTION xv 

CHAPTER  I. 

Art.    1.  Formulas  for  Work. — Temperature  Constant  ....  1 

Art.    2.  Formula  for  Work. — Temperature  Varying 3 

Art.    3.  Formula  for  Work. — Incomplete  Expansion 7 

Art.  3a.  Work  as  Shown  by  Indicator  Diagrams 8 

Art.    4.  Effect  of  Clearance. — In  Compression 9 

Art.    5.  Effect  of  Clearance  and  Compression  in  Expansion 

Engines 14 

Art.    6.  Effect  of  Heating  Air  as  it  Enters  Cylinders 17 

Art.    7.  Change  of  Temperature  in  Compression  or  Expansion  18 

Art.    8.  Density  at  Given  Temperature  and  Pressure    ....  19 

Art.  8a.  Weight  of  Moist-Air  .    .    .  •_,   v  ||f:  .   v 19 

Art.    9.  Cooling  Water  Required  <"  ..  .'   .' 'j  ..  \~ 20 

Art.  10.  Reheating  and  Cooling.    . -, 21 

Art.  11.  Compounding.    .*'....--.•   .-.;....    .....  23 

Art.  12.  Proportions  for  Compounding 25 

Art.  13.  Work  in  Compound  Compression 27 

Art.  14.  Work  under  Variable  Intake  Pressure    ...    L  ....  27 

Art.  15.  Exhaust  Pumps *".    ..:..,....  29 

Art.  16.  Efficiency  when  Air  is  Used  without  Expansion  ...  30 

Art.  17.  Variation  of  Free  Air  Pressure  with  Altitude    ....  31 

CHAPTER  II.     MEASUREMENT  OF  AIR. 

Art.  18.  General  Discussion 33 

Art.  19.  Apparatus  for  Measuring  Air 34 

Art.  20.  Measurement  by  Standard  Orifice 35 

Art.  21.  Formula — Standard  Orifice  under  Standard  Conditions  35 

Art.  22.  Apparatus  for  Measuring  Air  at  Atmospheric  Pressure  37 

Art.  23.  Coefficients  for  Large  Orifices 38 

Art.  24.  Apparatus  for   Measuring  Air  under  Pressure  with 

Standard  Orifice 41 

Art.  25.  Coefficients  and  Orifice  Diameters  for  Measurements 

at  High  Pressure 43 

Art.  26.  Discharge  of  Air  through  Orifices — Considerable  Drop 

in  Pressure 45 

Art.  27.  Air  Measurement  in  Tanks 46 

CHAPTER  III.     FRICTION  IN  AIR  PIPES. 

Art.  28,  General  Discussion 49 

Art.  2?).  The  Formula  for  Practice 49 

ix 


x  CONTENTS 

PAGE 

Art.  30.  Theoretically  Correct  Friction  Formula 57 

Art.  31.  Efficiency  of  Power  Transmission  by  Compressed  Air.  60 

CHAPTER  IV.     OTHER  AIR  COMPRESSORS. 

Art.  32.     Hydraulic  Air  Compressors — Displacement  Type  .    .  63 

Art.  33.  Hydraulic  Air  Compressors — Entanglement  Type  .    .  64 

Art.  34.  Centrifugal  or  Turbo-air  Compressors 66 

CHAPTER  V.     SPECIAL  APPLICATIONS  OF  COMPRESSED  AIR. 

Art.  35.  Return-air  System .;  .    ;   .  '    ...  68 

Art.  36.  Return-air  Pumping  System  .    .    .V.   '.,   .    .    .  ./..".  69 

Art.  37.  Simple  Displacement  Pump 75 

CHAPTER  VI.     THE  AIR-LIFT  PUMP. 

Art.  38.  General  Discussion "N.    .....    .    .    .  76 

Art.  39.  Theory  of  the  Air-lift  Pump  ....    r  ^ ../..    .  A    .  76 

Art.  40.  Design  of  Air-lift  Pumps.    .  '.    .    .  -.    . \.  C  •-  ;    ;    .  78 

Art.  41.  The  Air-lift  as  a  Dredge  Pump  .    .  ; .    .  '.  >   .  -'.    ,.    .  83 

Art.  42.  Testing  Wells  with  the  Air-lift  ....    .    .'  .   '.  V  .  84 

Art.  43.  Data  on  Operating  Air-lifts 85 

CHAPTER  VII.     RECEIVERS  AND  STORAGE  OF  COMPRESSED  AIR. 

Art.  44.  Receivers  for  Suppressing  Pulsations  Only 87 

Art.  45.  Receivers — Some  Storage  Capacity  Necessary.    ...  88 

Art.  46.  Hydrostatic — Compressed  Air  Reservoirs 89 

CHAPTER  VIII.    FANS. 

Art.  47.  Introductory *    .  91 

Art.  48.  Purely  Centrifugal  Effects 93 

Art.  49.  Impulsive  or  Dynamic  Effects 95 

Art.  50.  Discharging  Against  Back  Pressure 96 

Art.  51.  Designing 98 

Art.  52.  Testing 100 

Art.  53.  Suggestions 103 

CHAPTER  IX.     CENTRIFUGAL  OR  TURBO-AIR  COMPRESSORS. 

Art.  54.  Centrifugal  Compression  of  an  Elastic  Fluid    ....  105 

Art.  55.  Effect  of  Picking  up  the  Fluid 106 

Art.  56.  Working  Formula 107 

Art.  57.  Suggestions 110 

Art.  58.  Proportioning Ill 

CHAPTER  X.     ROTARY  BLOWERS. 

Art.  59.  General 113 

CHAPTER  XI.  EXAMPLES  AND  EXERCISES. 

Art.  60.  Introductory 115 

TABLES 127 

APPENDIX  A.     Drill  Capacity  Tables 169 

APPENDIX  B.     Design  of  Logarithmic  Charts 172 

APPENDIX  C.     Determination  of  Friction  Factors 179 

APPENDIX  D.     Oil  Differential  Gage 187 

INDEX  .  191 


SYMBOLS 

For  ready  reference  most  of  the  symbols  used  in  the  text  are  assembled 
and  denned  here. 

p  =  intensity  of  pressure  (absolute),  usually  in  pounds  per  square  foot. 
Compressed-air  formulas  are  much  simplified  by  using  pressures 
and  temperatures  measured  from  the  absolute  zero.  Hence 
where  ordinary  gage  pressures  are  given,  p  =  gage  pressure  + 
atmospheric  pressure.  In  the  majority  of  formulas  p  must  be 
in  pounds  per  square  foot,  while  gage  pressures  are  given  in 
pounds  per  square  inch.  Then  p  =  (gage  pressure  +  atmos- 
pheric pressure  in  pounds  per  square  inch)  X  144. 
v  =  volume — usually  in  cubic  feet. 

Where  sub-a  is  used,  thus  pa,  va,  the  symbol  refers  to  free  air 
conditions. 

higher  pressure 

r  =  ratio  of  compression  or  expansion  =  T~^  — 

lower  pressure 

The  lower  pressure  is  not  necessarily  that  of  the  atmosphere. 
t  =  absolute  temperature  =  Temp.  F.  +  460.6. 
n  —  an  empirical  exponent  varying  from  1  to  1.41. 
loge  =  hyperbolic  logarithm  =  (common  log.)  X  2.306. 
W  =  work — usually  in  foot-pounds. 
Q  =  weight  of  air  passed  in  unit  time. 
w  =  weight  of  a  cubic  unit  of  air. 
Other  symbols  are  explained  where  used. 


INDEX  TO  FORMULAS 

Number  Formula  Art.    PAGE 

1.  W  =  piVi  log,-  r;  isothermal  compression. . . 1  2 

2.  W  =  63.871  log™  r  for  one  pound  at  60° 1  2 

3.  2^1  =  Y ;  physical  law  of  gases ..'.. 2  3 

4.  pv  =  53.35 1  for  one  pound 2  3 

5.  p\vin  —  pxvxn  =  pzVz")  observed  law  of  gases 2  3 

6        w  =  p«t>t  -  Pi»i  +  ^2  _  p^ 2  4 

7-       ^  =  rT^I  (P2V2  ~~  Paya) 2  4 

7a.     TT  =  -  ^— r  53.35  (^  -  <i)  for  one  pound 2  4 

n  -  1 

fr    n      -  l) 2  5 

n-l 

8d.     TF  =  r  (—  ^-|J  53.35  (r   n      -  l\  1 1  =  fcf,  for  one  pound  2  6 

9        w  —  P^.    ^  PlVl  _j_  p2l,2  _  pai;i  |  incomplete  expansion ...  3  8 

10.       m.e.p.  =  2.3  pa  logio  r 4a  12 

10a.     m.e.p.  =  ^-^  Pa  (r    n      -  l\    4a  12 

lOb.     m.e.p.  =  p2  (kU  ~  f ")   -  p« 4a  12 


=  1  +  c\l  -rn  /  ; 


11.  E    =  1  +  c\l  -rn  /  ;  volumetric  efficiency  .............       46       13 

12.  tt    =  «!  =  -  <ir  "^~  .........................       7         18 


13c.     w    =  (m  -  0.38  Hq)  ;  moist  air  ..................       8a       20 

14.       d2   =  —  Jr  and  d3  =  —  ;  compounding  ...................     12         26 

ri>*  n 

n-  1 

15-       ^  =  ^~ZTi  (PBVfl)  (ri    "      -  lj  X  2;  compound  two-stage     13        27 

n-  1 

16.       IT  =  -  ^  pava  (n    n     -  Ij  X  3;  compound  three-stage     13        27 


xiv  INDEX  TO  FORMULAS 

Number                                                      Formula  Art.  PAGE 

17.  m   =       *°g  ?«  ~  log  P*      ;  exhaust  pump . .  15  30 

log  (V  +  v)  —  log  V 

18.  JE7    =  -T ;  efficiency  without  expansion 16  31 

19.  pa  =  .4912[w  -  0.0001  (F  -  32)] ;  pressure  by  barometer.  17  32 

20.  logio  pa  =  1.4687  —  122  4 (T  +  460) '  Pressure  at  elevation  17  32 


21.  Q  =  c  0.1639  d2  \l-  rpa]  orifice  measure 21  36 

21a.     wa  =  1.321  -  ;  weight  by  mercury  column 21  37 

22.  di   =  -y ;  orifice  under  pressure 25  44 

26.  /  =  c  -^  — ;  friction  in  pipes 29  50 

/  civ  2  \  W 

27.  d  =   (  — : —  I     ;  friction  in  pipes 29  51 

,      0.1025  lvat  f  .  ,. 

28.  /  =  'fiTsT^'*  friction  in  pipes 29  51 

30.       logio  Pz  =  logio  Pi  —  c2  ^  -£—}  friction  in  pipes 30  58 

U,    T  pa 

32.  E  =  ,        2;  efficiency  of  transmission 31  60 

33.  E  =  —  ^r ;  efficiency-hydraulic  air  compression 32  63 

34.  ^  =  ^      T^ — — :  air  lift  pump . .            40  80 

Q        Epa\oger' 

36.  ~  =  j!\  +  h  ;  air  lift  pump 40  80 

Q       35  logio  r' 

37.  H  =  —                  — ;  fans — centrifugal  pumps '49  96 

37a.     loge  RI  =  —  ( )  ;  turbo-air  compression  ....  55  107 

Pa  \  9  I 

38.  Rn  =  Rin  and  log  Rn  =  n  log  RI  ;  turbo-air  compression  . .  55  107 

•      logio  Ri  =   (2.3  X  53.3£ 

pression 56  108 


39.      lo,,  B,  =     2.3  x  53.35  x  ,  x       «2  -  ™>  t^bo-air  com- 


INTRODUCTION 

Compressed  Air  is  a  manufactured  product  of  considerably 
greater  value  than  the  things  used  and  consumed  in  its  manu- 
facture. The  things  used  in  its  production  are  power,  machinery, 
and  superintendence — chiefly  power.  Since  the  compressed  air 
is  then  more  costly  than  the  power  applied  in  its  production  it  is 
but  reasonable  that  we  should  give  as  much  attention  to  its 
efficient  use,  or  more,  than  we  would  to  the  use  of  steam,  water 
power  or  electric  energy.  Yet  in  much  of  the  practice  in  using 
compressed  air  this  anticipation  does  not  seem  to  be  realized. 
This  may  be  due  to  the  fact  that  the  exceeding  convenience  and 
safety  of  compressed  air,  and  its  labor-saving  qualities  in  many 
applications,  make  efficiency  as  measured  by  foot-pounds,  a 
secondary  consideration;  and  from  this  a  habit  of  wastefulness  is 
formed.  A  further  explanation  may  be  found  in  its  harmlessness 
and  general  good  nature.  A  leaky  air  pipe,  or  excessive  use  at  a 
motor,  does  not  scald,  suffocate,  nor  becloud  the  view  as  with 
steam;  nor  shock,  burn  nor  start  a  fire  as  with  electricity,  nor 
flood  and  foul  the  premises  as  with  water.  Hence  the  user  is  apt 
to  tolerate  wastes  of  compressed  air  that  should  be  checked  to 
save  the  coal  pile. 

In  some  lines  of  industry  compressed  air  is  supreme,  in  others 
electricity,  and  still  in  others  steam,  and  water,  each  being  specially 
adapted  to  do  certain  things  better  than  any  other,  but  in  some 
lines  the  winner  has  not  been  decided,  or  even  though  decided  in 
so  far  as  present  methods  apply  there  may,  any  day,  arise  fresh 
competition  by  the  invention  of  new  devices  or  processes. 


XV 


COMPRESSED  AIR 


CHAPTER  I 
f 

FORMULAS  FOR  WORK 

Art.  1.  Temperature    Constant  or   Isothermal   Conditions.— 

From  the  laws  of  physics  (Boyle's  Law)  we  know  that  while 
the  temperature  remains  unchanged  the  product  pv  remains 
constant  for  a  fixed  amount  (weight)  of  air.  Hence  to  determine 
the  work  done  on,  or  by,  air  confined  in  a  cylinder,  when  condi- 
tions are  changed  from  piv\  to  p&2  we  can  write 


PxVx   =   P2V2, 

sub  x  indicating  variable  intermediate  conditions. 


L 

p- 

PI-» 

• 

*-Px 

Lf 

6 

Lr 

\ 

b 

FIG.  1. 


Whence  px  =  —       and  dW  =  pxAdl  =  pxdvx  since  Adi  =  dv; 

Vx 

A    being   the   area   of   cylinder,    therefore   dW  =  piVi  —  -,  and 

Vx 

work  of  compression  or  expansion  between  points  B  and  C  (Fig. 
1)  is  the  integral  of  this,  or 


COMPRESSED  AIR 
W  =  p&i  I      -^  =  piVi  (log.  vi  -  log,  vz) 

Jv>      V* 
Vi  loge         =   piVi  loge  -       =   p&i  loge  f   =    p2V2  loge  7*. 


Note  that  this  analysis  is  only  for  the  work  against  the  front 
of  the  piston  while  passing  from  B  to  C.  To  get  the  work  done 
during  the  entire  stroke  of  piston  from  B  to  D  we  must  note  that 
throughout  the  stroke  (in  case  of  ordinary  compression)  air  is 
entering  behind  the  piston  and  following  it  up  with  pressure  pi. 
Note  also  that  after  the  piston  reaches  C  (at  which  time  valve  / 
opens)  the  pressure  in  front  is  constant  and  =  p2  for  the  remainder 
of  the  stroke.  Hence  the  complete  expression  for  work  done  by, 
or  against,  the  piston  is 


loge  r  -  piVi  +  p2v2-, 
but  since  p\v\  =  p2v2,  the  whole  work  done  is 

W  =  piVi  loge  r  or  p2vz  loge  r  (1) 

Note  that  the  operation  may  be  reversed  and  the  work  be  done 
by  the  air  against  the  piston,  as  in  a  compressed-air  engine,  with- 
out in  any  way  affecting  the  formula. 

Forestalling  Art.  2,  Ecj.  (4),  we  may  substitute  for  pv  in  Eq.  (1) 
its  equivalent,  53.35£,  for  1  Ib.  of  air  and  get  for  1  Ib.  of  dry  air 

W  =  53.35  *  Xloger  (la) 

This  may  be  adopted  for  common  logs  by  multiplying  by  2.3026. 
It  then  becomes 

W  =  (122.83  logior)*  (Ib) 

log  122.83  =  2.08930. 

Note  that  in  solving  by  logs  the  log  of  log  r  must  be  taken. 
Values  of  the  parenthesis  in  Eq.  (16)  are  given  in  Table  I, 
column  11. 

For  the  special  temperature  of  60°F.  (16)  becomes  for  1  Ib.  of 
air 

W  =  63,871  logio  r  (2) 

log  63,871  =  4.80536. 

Note  that  for  moist  air  the  coefficient  in  (la)  is  greater,  being 
£3.87  for  saturated  air  at  70°F.  and  under  atmospheric  pressure 


FORMULAS  FOR  WORK  3 

=  14.7.  For  average  conditions  53.5  would  probably  be  about 
right. 

Example  la.  —  What  will  be  the  work  in  foot-pounds  per  stroke 
done  by  an  air  compressor  displacing  2  cu.  ft.  per  stroke,  com- 
pressing from  pa  =  14  Ib.  per  square  inch  to  a  gage  pressure 
=  70  Ib.;  compression  isothermal,  T  =  60°F.? 

Solution  (a).  —  Inserting  the  specified  numerals  in  Eq.  (1)  it 
becomes 

W  =  144  X  14  X  2  X  loge  7°^U  =  4,032  X  1.79  =  7,217. 

Solution  (&).—  By  Tables  I  and  II. 

By  Table  II  the  weight  of  a  cubic  foot  of  air  at  14  Ib.  and  60° 
is  0.07277,  and  0.07277  X  2  =  0.14554.  The  absolute  t  is  460 
+  60  =  520,  and  r  =  6.0. 

Then  in  Table  I,  column  11,  opposite  r  =  6  we  find  95.271, 
whence 

W  =  95.271  X  520  X  0.14554  =  7,208. 

The  difference  in  the  two  results  is  due  to  dropping  off  the  fraction 
in  temperature. 

Art.  2.  Temperature  Varying.  —  The  conditions  are  said  to  be 
adiabatic  when,  during  compression  or  expansion,  no  heat  is  al- 
lowed to  enter  in,  or  escape  from,  the  air  although  the  temperature 
in  the  body  of  confined  air  changes  radically  during  the  process. 

Physicists  have  proved  that  under  adiabatic  conditions  the 
following  relations  hold  : 

/ON 


tz 

and  since  for  1  Ib.  of  air  at  32°F.  pv  =  26,214  and  t  =  492,  we 
get  for  1  Ib.  of  dry  air  at  any  pressure,  volume  and  temperature, 

pv  =  53.35*  (4) 

While  formulas  (3)  and  (4)  are  very  important,  they  do  not  apply 
to  the  actual  conditions  under  which  compressed  air  is  worked, 
for  in  practice  we  get  neither  isothermal  nor  adiabatic  conditions 
but  something  intermediate.  Furthermore,  moisture  in  the  air 
will  effect  this  coefficient. 

For  such  conditions  physicists  have  discovered  that  the  follow- 
ing holds  nearly  true  : 

g 

(5) 


4  COMPRESSED  AIR 

sub  x  indicating  any  intermediate  stage  and  the  exponent  n 
varying  between  1  and  1.41  according  to  the  effectiveness  of  the 
cooling  in  case  of  compression  or  the  heating  in  case  of  expan- 
sion. From  this  basic  formula  (5)  the  formulas  for  work  must 
be  derived. 

As  in  Art.  1,  dW  =  pxdvx  =  p\vf  —  *n  =  p\Vin  (vx~n)  dvx. 

vx 

Therefore 

w 


Now  since  p\v-^n  X  v2l~n  =  p2v2n  X  v2l~n  =  p2v2  and 
=  piVi  the  expression  becomes 


n  —  1 

which  represents  the  work  done  in  compression  or  expansion  be- 
tween B  and  C  (Fig.  1).  To  this  must  be  added  the  work  of  ex- 
pulsion, p2v2  and  from  it  must  be  subtracted  the  work  done  by 
air  against  the  back  side  of  the  piston.  In  case  of  compression 
from  free  air  this  subtraction  will  be  pava.  Hence,  the  net  work 
done  in  one  stroke  of  volume  va  is 


p.v.  (6) 

This  reduces  to 

W  = 


By  substituting  from  Eq.  (4);  Eq.  (7)  may  be  written,  for  work 
in  compressing  1  Ib.  of  air, 

Wi=  rz^-y53.35  (t2-ta)  (7a) 

Whenn  =  1.41,  Wi  =  183  (t2  -  ta)  (76) 

When  n  =  1.25,  Wi  =  266  (t2  -  ta)  (7c) 

Equation  (7)  applies  also  to  any  cases  of  complete  expansion, 

that  is,  when  the  air  is  expanded  until  the  pressure  within  the 

cylinder  equals  that  against  which  exhaust  must  escape. 

Equation  (7)  is  in  convenient  form  for  numerical  computations 
and  may  be  used  when  the  data  are  in  pressures  and  volumes,  but 
it  is  common  to  express  the  compression,  or  expansion,  in  terms  of 
r.  For  such  cases  a  more  convenient  form  of  equation  is  gotten 
as  follows: 


FORMULAS  FOR  WORK  5 

f*\    n\     n\    71  — 

From  Eq.  (5)  by  factoring  out  one  v,  p2v2  =       na_°  —  ' 

p2       van  va          1 

Also  r  =  —  =  —  -.  therefore  —  =  r  » 

pa       v2n)  v2 

n—l  re  —  1  re—  1 

and  °n_1  =  r  n  ,  therefore  p2v2  =  pava  r  n 

and  Eq.  (7)  becomes 

(8) 

In  cases  the  higher  pressure,  p2)  and  the  less  volume,  v2,  are 
known,  as  may  sometimes  be  the  case  in  complete  expansion  en- 
gines, we  would  get  by  a  similar  process 


Study  of  the  derivation  of  Eqs.  (7)  and  (8)  shows  that  they  are 
equally  applicable  to  cases  of  complete  expansion,  that  is,  when 
the  air  within  the  cylinder  is  expanded  until  its  pressure  is  equal 
to  that  of  the  air  outside  into  which  the  exhaust  takes  place. 
In  ordinary  cases  of  expansion  engines  apply  Eq.  (9). 

In  perfectly  adiabatic  conditions  n  =  1.41,  but  in  practice  the 
compressor  cylinders  are  water-jacketed  and  thereby  part  of  the 
heat  of  compression  is  conducted  away,  so  that  n  is  less  than  1.41. 
For  such  cases  Church  assumes  n  =  1.33  and  Unwin  assumes 
n  =  1.25.  Undoubtedly  the  value  varies  with  size  and  propor- 
tions of  cylinders,  details  of  water-jacketing,  temperature  of 
cooling  water  and  speed  of  compressors.  Hence  precision  in  the 
value  of  n  is  not  practicable. 

For  1  Ib.  of  air  at  initial  temperature  of  60°F.  Eq.  (8)  gives,  in 
foot-pounds, 

When  n  =  1.41,  W  =  95,193  (r0-29  -  1)  (86) 

When  n  =  1.25,  W  =  138,  405  (r0-2  -  1)  (8c) 

Common  log  of   95,193  =  4.978606. 

Common  log  of  138,405  =  5.141141. 

Values  of  r0-2  and  r0-29  are  given  in  Table  I,  columns  5  and  6 
respectively. 

The  above  special  values  will  be  found  convenient  for  approxi- 
mate computations.  For  compound  compression  see  Art.  14. 

If  in  Eq.  (8)  we  substitute  for  pv  its  value,  53.352,  for  1  Ib., 
we  get  for  work  on  1  Ib. 


COMPRESSED  AIR 

n-l 


where 


W  =  [(n^l)53<35(r    "      "1)]X^ 

-•) 


Kt 


(Sd) 


n-l 


X  53.35     r   n 


Note  that  Eq.  (8d)  applies  without  change  to  cases  of  complete 
expansion  provided  that  the  temperature,  h,  of  the  exhaust  be 
used  and  that  r  be  determined  to  correspond,  see  Eqs.  (12)  and 


Table  I  gives  values  of  K  for  n  =  1.25  and  n  =  1.41  and  for 
values  of  r  up  to  10,  varying  by  one-tenth.     The  theoretic  work 


FIG.  2. 

in  any  case  is  K  X  Q  X  t,  where  Q  is  the  number  of  pounds  passed 
and  t  is  the  absolute  initial  temperature.  Further  explanation 
accompanies  the  table. 

The  difference  between  isothermal  and  adiabatic  compression 
(and  expansion)  can  be  very  clearly  shown  graphically  as  in  Fig. 
2.  In  this  illustration  the  terminal  points  are  correctly  placed 
for  a  ratio  of  5  for  both  the  compression  and  expansion  curve. 

Note  that  in  the  compression  diagram  (a),  the  area  between  the 
two  curves  aef  represents  the  work  lost  in  compression  due  to 
heating,  and  the  area  between  the  two  curves  aeghb  in  (b)  repre- 


FORMULAS  FOR  WORK  7 

sents  the  work  lost  by  cooling  during  expansion.  The  isothermal 
curve,  ae,  will  be  the  same  in  the  two  cases. 

Such  illustrations  can  be  readily  adapted  to  show  the  effect  of 
reheating  before  expansion,  cooling  before  compression,  heating 
during  expansion,  etc.  For  platting  curves,  see  Art.  4a. 

Example  2a.  —  What  horsepower  will  be  required  to  compress 
1,000  cu.  ft.  of  free  air  per  minute  from  pa  =  14.5  to  a  gage  pres- 
sure =  80,  when  n  =  1.25  and  initial  temperature  =  50°F.? 

Solution.  —  From  Table  II,  interpolating  between  40° 
and  60°  the  weight  of  1  cu.  ft.  is  0.07686  and  the  weight  of 
1,000  is  76.86-.  The  r  from  above  data  is  6.5.  Then  in 
Table  I  opposite  r  =  6.5  in  column  9  we  find  0.3658.  Then 


Horsepower  =  0.3658  X 


X  510  =  143. 


The  student  should  check  this  result  by  Eqs.  (8)  or  (Sd)  and  (106) 
without  the  aid  of  the  table. 

Art.  3.  Incomplete  Expansion.  —  When  compressed  air  is  applied 
in  an  engine  as  a  motive  power  its  economical  use  requires  that  it 


FIG.  3. 

be  used  expansively  in  a  manner  similar  to  the  use  of  steam.  But 
it  is  never  practicable  to  expand  the  air  down  to  the  free  air  pres- 
sure, for  two  reasons :  first,  the  increase  of  volume  in  the  cylinders 
would  increase  both  cost  and  friction  more  than  could  be  balanced 
by  the  increase  in  power;  and  second,  unless  some  means  of  re- 
heating be  provided,  a  high  ratio  of  expansion  of  compressed  air 
will  cause  a  freezing  of  the  moisture  in  and  about  the  ports. 

The  ideal  indicator  diagram  for  incomplete  expansion  is  shown 
in  Fig.  3.  In  such  diagrams  it  is  convenient  and  simplifies  the 
demonstrations  to  let  the  horizontal  length  represent  volumes. 
In  any  cylinder  the  volumes  are  proportional  to  the  length. 


8  COMPRESSED  AIR 

Air  at  pressure  p2  is  admitted  through  that  part  of  the  stroke 
represented  by  v2 — thence  the  air  is  expanded  through  the  re- 
mainder of  the  stroke  represented  by  v\t  the  pressure  dropping 
to  pi.  At  this  point  the  exhaust  port  opens  and  the  pressure 
drops  to  that  of  the  free  air.  The  dotted  portion  would  be 
added  to  the  diagram  if  the  expansion  should  be  carried  down 
to  free  air  pressure. 

To  write  a  formula  for  the  work  done  by  the  air  in  such  a  case 
we  will  refer  to  Eq.  (6)  and  its  derivation.  In  the  case  of  simple 
compression  or  complete  expansion  it  is  correctly  written 


TIT  ~~   PaVa 

W   = 


which  would  give  work  in  the  case  represented  by  Fig.  1  when 
there  is  a  change  of  temperature,  but  in  such  a  case  as  is  repre- 
sented by  Fig.  3  the  equation  must  be  modified  thus: 


w  = 


the  reason  being  apparent  on.  inspection. 

In  numerical  problems  under  Eq.  (9)  there  will  be  known 
and  either  pi  or  vi.     The  unknown  must  be  computed  from  the 
relations  from  Eq.  (5)  : 


=     .  or  »i  =  »» 


Table  I,  columns  1,  2,  3  and  4,  is  designed  to  reduce  the  labor 
of  this  computation. 

Example  3a. — A  compressed-air  motor  takes  air  at  a  gage 
pressure  =  100  Ib.  and  works  with  a  cut-off  at  Y±  stroke.  What 
work  (foot-pounds)  will  be  gotten  per  cubic  foot  of  compressed 
air,  assuming  free  air  pressure  =  14.5  Ib.  and  n  =  1.41? 

Solution. — Applying  Eq.  (9)  and  noting  that  all  pressures  are  to 
be  multiplied  by  144  and  that  the  pressure  at  end  of  stroke 

/1X\  1.41 

=  pi  —.  114.5  (-^)         =  16.3    and     that     Vi  =  4v2,     we    get 


114.5  X  1  -  14.5  X4)  =  25,444. 


FORMULAS  FOR  WORK 


9 


Art.  4.  Work  "as  Shown  by  Indicator  Cards.—  Volumes  have 
been  written  on  indicators  and  indicator  cards  but  more  than  the 
following  brief  notes  would  be  out  of  place  here  : 

Let  h  =  length  in  inches  out  to  out,  horizontally,  of  indicator 

card, 

I    =  length  in  feet  of  piston  stroke, 
s    =  spring  number  =  pounds  per  inch, 
a    =  area  in  square  inches  of  indicator  diagram, 
A    =  area  in  square  inches  of  piston. 

Then  work  per  stroke  is 


W  =   T-  aAs 


(10) 


When  a  planimeter  ifc  available,  it  is  the  quickest  and  most 
reliable  means  of  determining  the  area,  a,  provided  the  operator 
know  the  planimeter  constant.  This  can  easily  be  found  by  run- 
ning the  planimeter  several  times  round  an  accurately  drawn 
figure  of  known  area,  as  for  instance  a  square  of  2-in.  sides,  and 
averaging  the  readings.  The  repetition  should  be  made  without 
lifting  the  tracer  from  the  paper.  It  is  necessary  only  to  read 
the  vernier  each  time  the  tracer  reaches  a  fixed  point  on  the 
figure.  Then  subtract  each  reading  from  the  next  succeeding  one. 
The  several  differences  reveal  whether  or  not  the  instrument,  in 
the  hands  of  the  individual  can  be  depended  on  to  give  reliable 
results.  The  sum  of  the  differences  divided  by  the  number  of 


Vernier 

Difference 

Error 

Per  cent. 

Reading  at  start 

794 

Pass  zero  

390 

1 

-0.25 

After  first  round 

184 

392 

After  second  round  

578 

394 

3 
392 

-0.75 

After  third  round 

964 

386 

5 

392 

-1.28 

Pass  zero  

398 

7 

-1.80 

After  fourth  round  

362 

392 

After  fifth  round  

753 

391 

0 
392 

0.00 

Total  for,  jive  rounds,  1,959 

-^— —  =  392  =  average  and  ^~^  =  1.03  =  planimeter  constant. 

O  o.  JZi 


10 


COMPRESSED  AIR 


repetitions  gives  the  average  and  the  average  divided  into  the 
known  area  gives  the  planimeter  constant. 

Example. — The  table,  page  9,  shows  records  and  deductions  for 
a  planimeter  tested  on  a  rectangle  of  4  sq.  in.:  '* 

To  get  the  full  information  shown  by  an  indicator  diagram 
taken  from  an  air  compressor,  there  should  be  placed  on  it  the 
clearance  line  and  the  isothermal  curve.  For  direct  determina- 
tion of  clearance  see  Art.  46. 


A    10 


i    D  o 


Referring  to  Fig.  4;  the  clearance  line,  OC,  is  placed  at  a  dis- 

DO 
tance,  DO,  such  that  -r;  =  percentage  of  clearance.     If  clear- 


ance has  been  determined  by  measurements  in  the  machine,  the 
line  OC  is  set  out  by  measuring  a  distance  CB  as  determined  above. 
If  clearance  has  not  been  measured  in  the  machine,  the  position 
of  the  line  OC  or  point  0  can  be  computed  as  follows  : 

Scale  one  of  the  pressure  ordinates  where  the  curve  is  smooth. 
Represent  this  by  px;  the  distance  from  its  foot  to  the  unknown 
point  0  represent  by  x  and  the  known  distance  from  A  to  the 
foot  of  px  by  k. 


FORMULAS  FOR  WORK  11 

Then  by  Eq.  (5) 

Pa  (x  +  k)n  =  pxxn  or  x  +  k  =  @$*x. 

\Pa 

k 
Whence  x  =—  /  N 


This  method  is  of  course  dependent  on  the  assumed  value  of  n. 

By  the  same  principle,  if  0  can  be  correctly  located  independ- 
ently of  n,  the  value  of  n  can  be  computed  thus:  x  is  now  known. 
So  let  x  +  k  =  L 

Then'  l-  =  *  and  »  -  lo.S  P!  ~  i°g  P"  (6) 

rcn       pa  log  I  —  log  z 

In  large  well-designed  air  compressors  the  clearance  should  not 
exceed  1  per  cent.  Then  OC  would  very  nearly  coincide  with 
DB  but  would  always  be  a  little  outside. 

Note  that  if  x  from  Eq.  (a)  places  0  inside  of  D,  it  is  evidence 
that  n  has  been  assumed  too  small. 

In  many  books  on  steam  engines  and  air  compressors  can  be 
found  instructions  for  locating  the  point  0  graphically.  Con- 
cerning these,  the  student  is  warned  that  they  are  all  based  on  the 
assumption  that  the  curve  is  the  isothermal;  and  hence  are  apt 
to  give  very  misleading  results.  For  instance,  one  method  is  to 
construct  a  rectangle  on  the  curve  as  developed  by  the  indicator, 
as  shown  in  mnqr,  and  the  diagonal  nr  will  pass  through  0. 
This  would  be  correct  for  the  isothermal  EF  but  evidently  not  so 
for  the  actual  curve  EG. 

Before  passing,  the  student's  attention  should  be  directed  to  the 
fact  that  in  steam  engines  the  clearance  is  usually  much  greater 
than  is  allowed  in  air  compressors.  In  steam  engines  it  varies 
much  and  sometimes  goes  to  10  or  even  15  per  cent.  ;  and  further, 
the  curves  of  steam-engine  indicator  cards  are  much  more  erratic 
than  for  air  compressors,  due  to  condensation  during  expansion 
and  compression  without  cooling,  which  may  cause  reevapora- 
tion. 

After  all  is  said,  if  the  investigator  wants  to  know  what  the 
clearance  is  he  should  measure  it  in  the  machine. 

The  curves,  either  isothermal  or  adiabatic,  can  most  readily 
be  set  out  by  dividing  the  line  OA  into  ten  equal  parts  numbered 
as  shown,  then  letting  x  =  number  of  divisions  from  Q  the  pres- 
sure ordinate  at  the  xth  division  will  be,  for  the  isothermal  curve  : 


12  COMPRESSED  AIR 


Wpa  /10\  '•« 

px  =  -    —  and  for  the  adiabatic  curve  px  =  pa  (  —  ) 
x  \  x  I 

The  following  table  applies  where  pa  =  14.7: 

x=        10       9876543          2 
Isothermal  p,  =      14.7  16.3  18.2  21.0  24.5  29.4  36.6  49.0     78.5 
Adiabatic  px  =      14.7  17.1  20.1  23.5  29.3  39.1  53.5  80.3  142.1 

The  isothermal  curve  is  symmetrical  about  the  middle  line 
and  the  upper  half  can  be  set  out  from  the  other  axis  OB  with  the 
same  ordinates  used  to  plat  the  first  half. 

Art.  4a.  Mean  Effect  Pressures.  —  In  much  of  the  literature 
relating  to  work  done  in  steam  engines  and  air  compressors,  use  is 
made  of  the  term  "mean  effective  pressure"  —  abbreviated  m.e.p. 
A  definition  of  the  term  is:  A  pressure  which  multiplied  by  the 
volume,  or  piston  displacement,  gives  work. 

Then  to  find  the  m.e.p.  in  compression  when  the  volume  is  v 
we  have: 

In  isothermal  compression  (or  expansion) 

(m.e.p.)^  =  PO.V  loge  r 

and  m.e.p.  =  pa  loge  r  =  2.3pa  logio  r.  (10) 

In  case  of  adiabatic  compression  (or  complete  expansion) 

n  /  n~l        \ 

(m.e.p.)v  =  -    —7  pav    (r  n    —  1) 

it  ~~  l  \  / 

and  m.e.p.  =  -    ^-y  pa  (r~^~   -  l\  (10a) 

Values  of  r~    -  are  given  in  Table  I,  column  5,  for  n  =  1.25. 

In  case  of  incomplete  expansion  (Eq.  9)  when  the  cutoff  is  at 
k  per  cent,  of  the  stroke,  or  v2  =  kv. 


f  s  . 

(m.e.p.)  v  =  -  -  _  /^     +  pzkv  —  pav. 

From  the  condition  that  p\vn  =  pzvzn,  we  get 

Pi  =  P 
whence  the  above  equation  reduces  to 


-  fc»)  (106) 

m.e.p.  =  — j— -  -  pa 

To  reduce  computations  of  m.e.p.  in  this  case  to  simple  arith- 


FORMULAS  FOR  WORK 


13 




metic,  the  values  of        __  ^   are  given  below  f  or  n  =  1.25  and  for 

a  sufficient  range  in  k  to  meet  the  demands  of  ordinary  practice. 
These  values  will  apply  to  any  gas,  including  steam  so  long  as 
there  is  no  condensation  of  the  steam. 


f  Fraction 
k] 
(  Decimal 

0.1250 

Me 
0.1875 

0.2500 

Me 
0.3125 

0.3750 

0.4375 

0.5000 

0.5625 

H 

0.6250 

kn  -  kn 

0.3287 

0.4439 

0.5428 

0.6281 

0.7006 

0.7643 

0.8180 

0.8641 

0.9022 

n-  I 

Art.  4b.  Effect  of  Clearance  in  Compression. — It  is  not  prac- 
ticable to  discharge  all  of  the  air  that  is  trapped  in  the  cylinder. 
There  are  some  pockets  about  the  valves  that  the  piston  cannot 
enter,  and  the  piston  must  not  be  allowed  to  strike  the  head  of  the 
cylinder.  This  clearance  can  usually  be  determined  by  measuring 
the  water  that  can  be  let  into  the  cylinder  in  front  of  the  piston 
when  at  the  end  of  its  stroke;  but  the  construction  of  each  com- 
pressor must  be  studied  before  this  can  be  undertaken  intelli- 
gently, and  it  is  not  done  with  equal  ease  in  all  machines. 

To  formulate  the  effect  of  this  clearance  in  the  operation  of 
the  machine, 

Let  v  =  volume  of  piston  displacement  (=  area  of  piston  X 
length  of  stroke), 

Let  cv  =  clearance,  c  being  a  percentage. 

Then  v  +  cv  is  the  volume  compressed  each  stroke.     But 

JL 

the  clearance  volume  cv  will  expand  to  a  volume  rncv  as  the 
piston  recedes,  so  that  the  fresh  air  taken  in  at  each  stroke  will 


be  v •+  cv  —  rncv,  and  the  volumetric  efficiency  will  be 


v  +  cv  —  rncv 


=  1  +  c  (1  -  rn) 


(ID 


Theoretically  (as  the  word  is  usually  used)  clearance  does  not 
cause  a  loss  of  work,  but  practically  it  does,  insomuch  as  it 
requires  a  larger  machine,  with  its  greater  friction,  to  do  a  given 
amount  of  effective  work. 

Example  46. — A  compressor  cylinder  is  12-in.  diameter  by 
16-in.  stroke.  The  clearance  is  found  to  hold  1J4  pt.  of  water 

1  25         *  ^fi 

X  231  =  36  cu.  in.,  therefore  c  =  ^,  ^  =  0.02. 


8 


113  X  16 


14 


COMPRESSED  AIR 


Then  by  Eq.  (11)  when  r  =  7  and  n  =  1.25. 

E  =  1  +  0.02(1  -  7°-8)  =  92  per  cent. 

Such  a  condition  is  not  abnormal  in  small  compressors,  and  the 
volumetric  efficiency  is  further  reduced  by  the  heating  of  air 
during  admission  as  considered  in  Art.  6. 

Art.  5.  Effect  of  Clearance  and  Compression  in  Expansion 
Engines. — Figure  5  is  an  ideal  indicator  diagram  illustrating  the 
effect  of  clearance  and  compression  in  an  expansion  engine. 

In  this  diagram  the  area  E  shows  the  effective  work,  D  the 
effect  of  clearance,  B  the  effect  of  back  pressure  of  the  atmosphere 
and  C  the  effect  of  compression  on  the  return  stroke. 

The  study  of  effect  of  clearance  in  an  expansion  engine  differs 
from  the  study  of  that  in  compression,  due  to  the  fact  that  the 


FIG.  5. 

volume  in  the  clearance  space  is  exhausted  into  the  atmosphere 
at  the  end  of  each  stroke. 

If  the  engine  takes  full  pressure  throughout  the  stroke  the  air 
(or  steam)  in  the  clearance  is  entirely  wasted ;  but  when  the  air  is 
allowed  to  expand  as  illustrated  in  the  diagram  some'  useful  work 
is  gotten  out  of  the  air  in  the  clearance  during  the  expansion. 

The  loss  due  to  clearance  in  such  engine  is  modified  by  the 
amount  of  compression  allowed  in  the  back  stroke.  If  the  com- 
pression pc  =  p2,  the  loss  of  work  due  to  clearance  will  be  noth- 
ing, but  the  effective  work  of  the  engine  will  be  considerably 
reduced,  as  will  be  apparent  by  a  study  of  a  diagram  modified  to 
conform  to  the  assumption. 

While  the  formula  for  work  that  includes  the  effect  of  clear- 
ance and  compression  will  not  be  often  used  in  practice  its  deriva- 
tion is  instructive  and  gives  a  clear  insight  into  these  effects. 


FORMULAS  FOR  WORK  15 

The  symbols  are  placed  on  the  diagram  and  will  not  need  fur- 
ther definition. 

The  effective  work  E  will  be  gotten  by  subtracting  from  the 
whole  area  the  separate  areas  B,  C  and  D.  From  Art.  2,  after 
making  the  proper  substitutions  for  the  volumes,  there  results 

Total 


Area  B  =  lpa, 
Area  D  = 


Subtracting  the  last  three  from  the  first  and   reducing   their 
results : 

~3T~  =  n^l  'C  ^2  +  Pa  ~  Pc  ~  Pl^  +  U  ^Pzk  +  Pab~pa)  ~  (pl 
—  pa)]  =  mean  effective  pressure. 
The  actual  volume  ratio  before  and  after  expansion  is 

vz  _  cli  +  kh  _  c  •+•  kt 

Vi  di  +  LI          C  +  1 

This  is  the  ratio  with  which  to  enter  Table  I  to  get  r  and  t  and 
from  r  the  unknown  pressure  p\.     Similarly,  for  the  compression 

f* 

curve  the  ratio  of  volumes  is  ^,  and  pc  can  be  found  as  indicated 

above. 

Art.  5a.  Adjustment  of  Mechanically  Operated  Intake 
Valves. — While  no  attempt  will  be  made  to  show  details  of  valves, 
it  is  appropriate  to  call  attention  here  to  the  fact  that  discharge 
valves  to  an  air  compressor  are  nearly  always  of  the  " poppet" 
type,  that  is  they  pop  open  automatically  when  the  pressure  in- 
side the  compressor  exceeds  that  in  the  receiver.  Thus  the  time, 
or  point  of  opening  of  the  discharge  valve,  will  adjust  itself  to  any 
variation  of  pressure  in  the  receiver,  a  condition  evidently  desir- 
able. But  the  intake  valves  may  be  mechanically  operated,  and 
are  so  operated  in  many  of  the  larger  machines.  When  so  oper- 
ated, the  machine  works  more  efficiently,  with  less  noise  and  there 
is  less  liability  to  breakdown. 

The  correct  adjustment  of  the  point  of  opening  of  mechanically 
operated  inlet  valves  depends  on  the  clearance  and  the  ratio  of 
compression. 


16 


COMPRESSED  AIR 


Figure  5a  illustrates  one  class  of  inlet  valve  with  its  operating 
mechanism,  the  direction  of  motion  being  indicated  by  arrows. 
The  piston  d  is  at  the  left  end  of  its  stroke  with  compressed  air 
in  the  clearance.  If  the  port  of  valve  A  opens  at  this  instant,  the 
compressed  air  in  the  clearance  will  escape  out  into  the  inlet 
passage  e.  Evidently  it  is  desirable  to  delay  the  opening  of  the 
port  until  the  piston  has  receded  enough  to  allow  the  air  in  the 
clearance  to  expand  down  to  atmospheric  pressure,  thus  letting 
the  air  in  the  clearance  give  back  the  work  done  in  compressing 
it.  Evidently,  the  opening  should  not  be  delayed  longer,  for 
there  would  result  a  suction  (pressure  below  atmosphere)  behind 
the  piston  which  would  cause  a  loss  of  work.  When  the  adjust- 
ment is  correct,  there  is  no  puffing  or  spitting  at  the  inlet  parts. 
When  the  adjustments  are  not  correct,  an  experienced  operator 
can  detect  the  fact  by  the  noise  made  by  air  puffing  out  or  into 
the  parts. 


SECTION    MIST 


SECTION  QR 


FIG.  5a. 


If  the  clearance  and  the  ratio  of  compression  are  known,  the 
erector  or  operator  can  adjust  the  valves  correctly.  For  example, 
assume  the  clearance  as  1  per  cent.,  r  =  7  and  n  =  1.25.  In 
Table  I  for  r  =  7,  v2  +  vi  =  0.21  or  say  Vi  =  5v2,  that  is  the  clear- 
ance should  expand  to  five  times  its  volume  before  the  port  opens. 
Otherwise  stated,  the  piston  should  move  back  4  per  cent,  of  its 
stroke  before  the  port  opens.  Thus,  if  the  stroke  be  18  in.  the 
piston  should  be  moved  back  0.04  X  18  =  0.72  (or  say  %  in.) 
and  while  the  piston  stands  in  that  position  bring  the  edge  of 
valve  and  edge  of  inlet  port  to  coincide  by  turning  the  rod  b  or  a 
as  the  case  may  be.  The  manufacturers  always  put  marks  on 
the  end  of  the  valve  and  on  the  inclosing  cylinder  that  will  enable 
the  operator  to  make  this  adjustment. 

In  order  that  one  adjustment  may  not  interfere  with  another, 
it  is  necessary  that  the  valve  B  be  adjusted  first  by  rotating 


FORMULAS  FOR  WORK  17 

rod  a;  then  adjust  valve  A  by  rotating  rod  b.  If  the  compressor 
be  a  compound  tandem,  adjust  the  valves  in  the  order  of  their 
distance  from  the  eccentric. 

Art.  6.  Effect  of  Heating  Air  as  it  Enters  Cylinders.  —  When 
a  compressor  is  in  operation  all  the  metal  exposed  to  the  compressed 
air  becomes  hot  even  though  the  water-jacketing  is  of  the  best. 
The  entering  air  comes  into  contact  with  the  admission  valves, 
cylinder  head  and  walls  and  the  piston  head  and  piston  rod,  and 
is  thereby  heated  to  a  very  considerable  degree.  In  being  so 
heated  the  volume  is  increased  in  direct  proportion  to  the  absolute 
temperature  (see  Eq.  (3)),  since  the  pressure  may  be  assumed 
constant  and  equal  that  of  the  atmosphere.  Hence  a  volume  of 
cool  free  air  less  than  the  cylinder  volume  will  fill  it  when  heated. 
This  condition  is  expressed  by  the  ratio 

Va         ta  ta 

-.=        or    Va  =  „-, 

Vc         lc  lc 

r~^     "*" 

where  vc  and  te  represent  the  cylinder  volume  and  temperature. 
The  volumetric  efficiency  as  effected  by  the  heating  is 


Example  6.  —  Suppose  in  Ex.  4a  the  outside  free  air  tempera- 
ture is  60°F.  and  in  entering  the  temperature  rises  to  160°F., 
then 

ta       460  +  60 

Tc  =  460+160  =  84  P 

Then  the  final  volumetric  efficiency  would  be  92  X  84  =  77 
per  cent,  nearly. 

The  volumetric  efficiency  of  a  compressor  may  be  further  re- 
duced by  leaky  valves  and  piston. 

In  Arts.  46  and  6  it  is  made  evident  that  the  volumetric  efficiency 
of  an  air  compressor  is  a  matter  that  cannot  be  neglected  in  any 
case  where  an  installation  is  to  be  intelligently  proportioned.  It 
should  be  noted  that  the  volumetric  efficiency  varies  with  the 
various  makes  and  sizes  of  compressors  and  that  the  catalog 
volume  rating  is  always  based  on  the  piston  displacement. 

These  facts  lead  to  the  conclusion  that  much  of  the  uncertainty 
of  computations  in  compressed-air  problems  and  the  conflict- 
ing data  recorded  is  due  to  the  failure  to  determine  the  actual 


18  COMPRESSED  AIR 

amount  of  air  involved  either  in  terms  of  net  volume  and  tempera- 
ture or  in  pounds. 

Methods  of  determining  volumetric  efficiency  of  air  compres- 
sors are  given  in  Chapter  II. 

The  loss  of  work  due  to  the  air  heating  as  it  enters  the  com- 
pressor cylinder  is  in  direct  proportion  to  the  loss  of  volumetric 
efficiency  due  to  this  cause.  In  Ex.  6a  only  84  per  cent,  of  the 
work  done  on  the  air  is  effective. 

By  the  same  law  any  cooling  of  the  air  before  entering  the 
compressor  effects  a  saving  of  power.  See  Art.  10. 

Art.  7.  Change  of  Temperature  in  Compression  or  Expan- 
sion. —  Equation  (4)  may  be  written  for  any  fixed  weight  of  air 


=   Ct2 

and  Eq.  (5)  may  be  factored  thus, 

Pivivf-1  =  p2v2v2n-1. 
Substituting  we  get 

ctivf-1  =  ct2vzn~l. 

Whence  h  =  *iW  (12) 


and  *2  =  =  jir,  (12a) 

i 
since  from  Eq.  (5)  -  =  (^-2)  w- 

t'2  \PI/ 

It  is  possible  to  compute  n  from  Eq.  (12)  by  controlling  the  Vi  and 
v2  and  measuring  ti  and  t2. 

Table  I,  columns  5  and  6,  is  made  up  from  Eq.  (12a)  and 
columns  3  and  4  from  Eq.  (5)  as  just  written. 

Example  7.  —  What  would  be  the  temperature  of  air  at  the  end 
of  stroke  when  r  =  7  and  initial  temperature  =  70°F.  ? 

Solution.  —  Referring  to  Table  I  in  line  with  r  =  7  note  that 

1.4758  when  n  =  1.25 


.-.  t2  =  (460  -f  70)  X  1.4758  -  460  =  322°F. 
1.7585  when  n  =  1.41 


/.  *2  =  (460  +  70)  X  1.7585  -  460  =  472°F. 

From  the  same  table  the  volume  of  1  cu.  ft.  of  free  air  when 
compressed  and  still  hot  would  be  respectively  0.21  and  0.25, 


FORMULAS  FOR  WORK  19 

while  when  the  compressed  air  is  cooled  back  to  70°  its  volume 
would  be  0.143. 

Art.  8.  Density  at  Given  Temperature  and  Pressure.  —  By  Eq. 
(4)  pv  =  53.35  for  1  lb.,  and  the  weight  of  1  cu.  ft.  =  1  Ib.  divided 
by  the  volume  of  1  lb. 

Therefore  w  =  l  =  7^7  (13) 

v       53.352 

Note  that  p  must  be  the  absolute  pressure  in  pounds  per  square 
foot,  and  t  the  absolute  temperature.  When  gage  pressures  are 
used  and  ordinary  Fahrenheit  temperature  the  formula  becomes 


/  ft_+p.\ 
53.35  \460  +  Ft 

(i3a) 


Table  III  is  made  up  from  Eq.  (13). 

Art.  8a.  Weight  of  Moist  Air.  —  In  some  cases  where  unusual 
refinement  of  calculations  may  be  required  it  will  be  necessary  to 
take  cognizance  of  the  fact  that  air  containing  water  vapor  is  not 
equal  in  weight  to  pure  air  at  the  same  pressure  and  temperature. 
In  case  of  moving  air  at  atmospheric  pressure  as  in  case  of  fans 
and  blowers  and  where  air  is  measured  at  atmospheric  pressure  by 
means  of  orifices,  the  error  resulting  from  neglecting  moisture  may 
be  as  much  as  1%  per  cent. 

Since   atmospheric   pressure   and  water-vapor  pressures  are 

usually  recorded  in  inches  of  mercury  it  will  be  convenient  to  re- 

tain in  the  formulas  inches  head  of  mercury  instead  of  pounds  per 

square  inch. 

.  Let   m  =  pressure  (absolute)  of  mixture  of  air  and  water  vapor 

in  inches  of  mercury, 

q  =  pressure  of  saturated  water  vapor  at  given  tempera- 
ture in  inches  of  mercury  (to  be  found  in  steam 
tables), 

H  =  percentage  of  humidity, 
K  =  ratio  of  weight  of  water  vapor  to  dry  air  at  given 

temperature, 
t  =  absolute  temperature  =  459.6  +  F. 

To  adopt  Eq.  (13),  viz.,  Wa  =  p  +  53.35  1  to  this  case,  note 
that  2.036  in.  of  mercury  gives  1  lb.  pressure  and  that.Hq  (  = 


20  COMPRESSED  AIR  ^ 

vapor  pressure  at  H  humidity)  must  be  subtracted  from  p  in  order 
to  get  the  true  pressure  of  the  air. 

.    144  (m  -  Hq)   ._  1.3253  (m  -  Hq) 
~  2.036  X  53.35*  "  t 

and  weight  of  water  vapor  in  a  cubic  foot  is 

1.3253  ,-„ 
ww  =  — - — KHq. 

Then  the  combined  weight  is 

wa  +  ww  =  w  =  -       —  [m  —  Hq  (1  —  K)]        (13&) 


For  values  of  K  see  Table  Ilia,  page  134,  which  is  copied  from 
Engineering  News,  June  18,  1908,  or  Compressed  Air  Magazine, 
vol.  13,  p.  4967.  These  articles  give  also  a  very  full  treatment 
of  the  subject  of  moisture  in  air. 

The  ratio  K  varies  between  0.611  at  zero  degrees  F.  and  0.623 
at  100°F.  Some  writers  assume  it  constant.  If  we  assume  it 
constant  and  equal  0.62  (which  is  correct  for  temperature  74°), 
then  the  equation  becomes 


w  =  (m  _  o.38#g)  (13c) 

£ 

Example  13c.  —  Find  the  weight  per  cubic  foot  of  air  in  a  duct 
leading  away  from  a  fan  when  T  =  70°F.  Barometer  reading 
in  free  air  =  28.85-in.-water  gage,  (i),  =  4  in.  and  humidity,  (H), 
=  80  per  cent. 

Solution.  —  4  in.  of  water  =  0.29  in.  of  mercury.  Then  m 
=  29.14. 

At  70°  K  =  0.6196  and  1  -  K  =  0.3804. 

At  70°  q  =  0.739  and  Hq  =  0.5912. 

I 


Then  w  =  ~~  (29.14  -  0.5912  X  0.3804)  =  0.07233. 

OoU 

Pure  air  under  the  same  pressure  and  temperature  would  have 
wa  =  0.07287,  a  difference  of  less  than  1  per  cent.  If  the  air  were 
saturated  the  difference  would  be  greater. 

Art.  9.  Cooling  Water  Required.  —  In  isothermal  changes,  since 
pv  is  constant,  evidently  there  is  no  change  in  the  mechanical 
energy  in  the  body  of  air  as  measured  by  the  absolute  pressure  and 
using  the  term  "  mechanical  energy"  to  distinguish  from  heat 
energy.  Hence  evidently  all  the  work  delivered  to  the  air  from 


FORMULAS  FOR  WORK  21 

outside  must  be  abstracted  from  the  air  in  some  other  form,  and 
we  find  it  in  the  heat  absorbed  by  the  cooling  water.     Therefore, 


of  work  must  be  absorbed  by  the  cooling  water.  If  the  water 
is  to  have  a  rise  of  temperature  T°  (T  being  small,  else  the 
assumption  of  isothermal  changes  will  not  hold),  then 

^  °g*r  =  pounds  of  water  required  in  same  time. 

/oU  1 

Example.  9  —  How  many  cubic  feet  of  water  per  minute  will  be 
required  to  cool  1,000  cu.  ft.  of  free  air  per  minute,  air  compressed 
from  pa  =  14.2  to  pg  =  90°  gage,  initial  temperature  of  air  = 
50°F.  and  rise  in  temperature  of  cooling  water  =  25°? 

Solution.  — 


144  X  14.2  X  1,000  X  log. 


"  =  3'36     CU'     ft" 


780  X  25  X  62.5 

minute. 

It  is  practically  possible  to  attain  nearly  isothermal  conditions 
by  spraying  cool  water  into  the  cylinder  during  compression.  In 
such  a  case  this  article  would  enable  the  designer  to  compute  the 
quantity  of  water  necessary  and  therefrom  the  sizes  of  pipes, 
pumps,  valves,  etc. 

Art.  10.  Reheating  and  Cooling.  —  In  any  two  cases  of  change 
of  state  of  a  given  weight  of  air,  assuming  the  ratio  of  change  in 
pressure  to  be  the  same,  the  work  done  (in  compression  or  expan- 
sion) will  be  directly  proportional  to  the  volume,  as  will  be  evident 
by  examination  of  the  formulas  for  work.  Also,  at  any  given 
pressure  the  volumes  will  be  directly  proportional  to  the  absolute 
temperatures.  Hence  the  work  done  either  in  compression 
or  expansion  (ratio  of  change  in  pressures  being  the  same  in 
each  case)  will  be  directly  p1  portional  to  the  absolute  initial 
temperatures. 

Thus  if  the  temperature  of  the  air  in  the  intake  end  of  one  com- 
pressor is  150°F.  and,  in  another  50°F.,  the  work  done  on  equal 
weights  of  air  in  the  two  cases  will  be  in  the  proportion  of  460  + 
150  to  460  +  50,  or  1.2  to  1;  that  is,  the  work  in  the  first  case  is 
20  per  cenir.  more  than  that  in  the  second  case.  This  is  equally 
true,  of  course,  for  expansion. 


22  COMPRESSED  AIR 

The  facts  above  stated  reveal  a  possible  and  quite  practicable 
means  of  great  economy  of  power  in  compressing  air  and  in  using 
compressed  air. 

The  opportunities  for  economy  by  cooling  for  compression  are 
not  as  good  as  in  heating  before  the  application  in  a  motor,  but 
even  in  compression  marked  economy  can  be  gotten  at  almost 
no  cost  by  admitting  air  to  the  compressor  from  the  coolest  con- 
venient source,  and  by  the  most  thorough  water-jacketing  with 
the  coolest  water  that  can  be  conveniently  obtained. 

In  all  properly  designed  compressor  installations  the  air  is 
supplied  to  the  machine  through  a  pipe  from  outside  the  building 
to  avoid  the  warm  air  of  the  engine  room.  In  winter  the  differ- 
ence in  temperature  may  exceed  100°,  and  this  simple  device 
would  reduce  the  work  of  compression  by  about  20  per  cent. 


Hot  Air 


Gas,  Liquid  or 
Powdered  Fuel 

Cold  Air/ 

FlG.   6. 

For  the  effect  of  inter  coolers  and  interheaters  see  Art.  11  on 
compounding. 

By  reheating  before  admitting  air  to  a  compressed-air  engine 
of  any  kind  the  possibilities  of  effecting  economy  of  power  are 
greater  than  in  cooling  for  compression,  the  reason  being  that 
heating  devices  are  simpler  and  less  costly  than  any  means  of 
cooling  other  than  those  cited  above. 

The  compressed  air  passing  to  an  engine  can  be  heated  to  any 
desired  temperature;  the  only  limit  is  that  temperature  that  will 
destroy  the  lubrication.  Suppose  the  normal  temperature  of  the 
air  in  the  pipe  system  is  60°F.  and  that  this  is  heated  to  300°F. 
before  entering  the  air  engine,  then  the  power  is  increased  46 
per  cent.  Reheating  has  the  further  advantage  that  it  makes 
possible  a  greater  ratio  of  expansion  without  the  temperature 
reaching  freezing  point. 

The  devices  for  reheating  are  usually  a  coil  or  cluster  of  pipes 
through  which  the  air  passes  while  the  pipe  is  exposed  to  the  heat 


FORMULAS  FOR  WORK  23 

of  combustion  from  outside.  Ordinary  steam  boilers  may  be 
used,  the  air  taking  the  place  of  the  steam  and  water. 

Unwin  suggests  reheating  the  air  by  burning  the  fuel  in  the 
compressed  air  as  suggested  in  the  cut. 

Even  when  the  details  are  worked  out  such  a  device  would  be 
simple  and  inexpensive.  The  theoretic  advantages  of  such  a 
device  are  that  all  the  heat  would  go  into  the  air,  the  gases  of  com- 
bustion (if  solid  or  liquid  fuel  be  used)  would  increase  the  volume, 
and  the  combustion  occurring  in  compressed  air  would  be  very 
complete. 

The  author  has  no  knowledge  of  any  such  devices  having  been 
used  in  practice.1 

The  power  efficiency  of  the  fuel  used  in  reheaters  is  very  much 
greater  than  that  of  the  fuel  used  in  steam  boilers.  Unwin  states 
that  it  is  five  or  six  times  as  much.  The  chief  reason  is  that  none 
of  the  heat  is  absorbed  in  evaporation  as  in  a  steam  boiler. 

In  many  of  the  applications  of  compressed  air  reheating  is  im- 
practicable, and  efficiency  is  secondary  to  convenience — but  in 
large  fixed  installations,  such  as  mine  pumps,  reheating  should  be 
applied. 

Art.  11.  Compounding. — In  steam-engine  designs  compound- 
ing is  resorted  to  to  economize  power  by  saving  steam,  while  in  air 
compressors  and  compressed-air  engines  compounding  is  resorted 
to  for  the  twofold  purpose  of  economizing  power  and  controlling 
temperature,  both  objects  being  accomplished  by  reducing  the 
extreme  change  of  temperature.  The  economic  principles  in- 
volved in  compound  steam  engines  and  in  compound  air  engines 
are  quite  different,  the  reasons  underlying  the  latter  being  much 
more  definite. 

The  air  is  first  compressed  to  a  moderate  ratio  in  the  low- 
pressure  cylinder,  whence  it  is  discharged  into  the  "intercooler," 
where  most  of  the  heat  developed  in  the  first  stage  is  absorbed  and 
thereby  the  volume  materially  reduced,  so  that  in  the  second 
stage  there  will  be  less  volume  to  compress  and  a  less  injurious 
temperature. 

The  changes  occurring  and  the  manner  in  which  economy  is 

1  Since  the  publication  of  the  first  edition  a  very  promising  device  has 
appeared  in  which  the  current  of  compressed  air  automatically  injects  the 
fuel  oil;  thus,  presumably,  maintaining  a  constant  proportion  between  the 
quantity  of-  air  and  of  oil,  so  that  the  temperature  of  the  discharged  air  will 
be  constant. 


24 


COMPRESSED  AIR 


effected  in  compression  may  be  most  easily  understood  by  refer- 
ence to  Fig.  7,  which  represents  ideal  indicator  diagrams  from 
the  two  cylinders,  superimposed  one  over  the  other,  the  scale 
being  the  same  in  each,  the  dividing  line  being  kb. 

In  this  diagram, 
abk  is   the  compression   line  in   the  first-stage  or   low-pressure 

cylinder, 
cds  is  the  compression  line  in  the  second-stage  or  high-pressure 

cylinder, 
be  is  the  reduction  of  volume  in  the  intercooler,  with  pressure 

constant, 


FIG.  7. 

abf  would  be  the  pressure  line  if  no  intercooling  occurred, 
The  area  cdfb  is  the  work  saved  by  the  intercooler, 
ace  would  be  the  compression  line  for  isothermal  compression, 
aug  would  be  the  compression  line  for  adiabatic  compression. 

The  diagram  is  correctly  proportioned  for  r  =  6. 

Figure  8  is  a  diagram  drawn  in  a  manner  similar  to  that  used 
in  Fig.  7  and  is  to  illustrate  the  changes  and  economy  effected 
by  compounding  with  heating  when  compressed  air  is  applied  in 
an  engine.  It  is  assumed  that  the  air  is  "preheated,"  that  is, 
heated  once  before  entering  the  high-pressure  cylinder  and  again 
heated  between  the  two  cylinders. 


FORMULAS  FOR  WORK 


25 


In  this  diagram, 

se  is  the  volume  of  compressed  air  at  normal  temperature, 
sf  is  the  volume  of  compressed  air  after  preheating, 
fc  is  the  expansion  line  in  the  high-pressure  cylinder, 
cb  is  the  increase  of  volume  in  the  interheater, 
by  is  the  expansion  line  in  low-pressure  cylinder, 
ezq  would  be  the  adiabatic  expansion  line  without  any  heating, 
efcz  is  work  gained  by  preheating, 
cbyx  is  work  gained  by  interheating. 

In  no  case  is  it  economical  to  expand  down  to  atmospheric 
pressure.  Hence  the  diagram  is  shown  cut  off  with  pressure  still 
above  that  of  free  air. 


e       f 


FIG.  8. 


The  diagram,  Fig.  8,  is  proportioned  for  preheating  and  reheat- 
ing 300°F. 

Art.  12.  Proportions  for  Compounding. — It  is  desirable  that 
equal  work  be  done  in  each  stage  of  compounding.  If  this  con- 
dition be  imposed,  Eq.  (8)  indicates  that  the  r  must  be  the  same 
in  each  stage,  for  on  the  assumption  of  complete  intercooling  the 
product  pv  will  be  the  same  at  the  beginning  of  each  stage. 

If  then  ri  be  the  ratio  of  compression  in  the  first  stage,  the 
pressure  at  end  of  first  stage  will  be  ripa  =  pi}  and  the  pressure  at 
end  of  seqond  stage  =  ripi  =  rizpa  =  p2,  and  similarly  at  end 
of  third  stage  the  pressure  will  be  p3  =  ri*pa)  or 


26  COMPRESSED  AIR 

In  two-stage  work  r\  =  l—\  **  =  r^' 

In  three-stage  work  ri  =  f^*jH  =  r^' 

Let  Vi  =  free  air  intake  per  stroke  in  low-pressure  cylinder  or  first 

stage, 

v2  =  piston  displacement  in  second  stage, 
t>3  =  piston  displacement  in  third  stage, 
7*1  =  ratio  of  compression  in  each  cylinder. 

Then,  assuming  complete  intercooling, 


=  —  and  vs  =  —  = 
TI  TI 


or 


Vi       ri         vi       ri2 

The  length  of  stroke  will  be  the  same  in  each  cylinder;  there- 
fore the  volumes  are  in  the  ratio  of  the  squares  of  diameters,  or 

dz2       1        ,  c?32        1 
-T-g  =  —  and  -T-J  =  ~Y 

Hence 

dz  =  — TZ  and  ds  =  -  (14) 

fi*  TI 

If  the  intention  to  make  the  work  equal  in  the  different  cylin- 
ders be  strictly  carried  out  it  will  be  necessary  to  make  the  first- 
stage  cylinder  enough  larger  to  counteract  the  effect  of  volu- 
metric efficiency.  Thus  if  volumetric  efficiency  be  75  per  cent., 
the  volume  (or  area)  of  the  intake  cylinder  should  be  one-third 
larger.  Note  that  the  volumetric  efficiency  is  chargeable 
entirely  to  the  intake  or  low-pressure  cylinder.  Once  the  air  is 
caught  in  that  cylinder  it  must  go  on. 

Example  12. — Proportion  the  cylinders  of  a  compound  two- 
stage  compressor  to  deliver  300  cu.  ft.  of  free  air  per  minute  at  a 
gage  pressure  =  150.  Free  air  pressure  =  14.0,  r.p.m.  =  100, 
stroke  18  in.,  piston  rod  1%  in.  diameter,  volumetric  efficiency 
=  75  per  cent. 

Solution. — From  the  above  data  the  net  intake  must  be  3  cu.  ft. 
per  revolution.  Add  to  this  the  volume  of  one  piston  rod  stroke 
( =  0.025  cu.  ft.),  and  divide  by  2.  This  gives  the  volume  of  one 
piston  stroke  1.512.  The  volume  of  1  ft.  of  the  cylinder  will  be 


FORMULAS  FOR  WORK  27 

12 

^  X  1.512  =  1.008  cu.  ft.     From  Table  X  the  nearest  cylin- 

lo 

der  is  14  in.   in   diameter,   the  total  ratio  of   compression    = 
-—  2  --  =  11.71,  and  the   ratio  in  each  stage   is  (11.71)**  = 
3.7  =  n,  and  by  (14) 

=  7.3  in.,  say  1%  in., 


for  the  high-pressure  cylinder. 

Now  we  must  increase  the  low-pressure  cylinder  by  one-third  to 
allow  for  volumetric  efficiency.  The  volume  per  foot  will  then 
be  1.344,  which  will  require  a  cylinder  about  15%  in.  in  diame- 
ter. Note  that  the  diameter  of  the  high-pressure  cylinder  will 
not  be  affected  by  the  volumetric  efficiency. 

Art.  13.  Work  in  Compound  Compression.  —  Assuming  that 
the  work  is  the  same  in  each  stage,  Eq.  (8)  can  be  adapted  to  the 
case  of  multistage  compression  thus: 

In  two-stage  work 

-l     X2  (15) 


n  —  1 

..  i     X2.  (15a) 


In  three-stage  work 

n-l 

W  =  ^—  Pava  (n  n        -  l)  X  3  (16) 

n-l 

3  (16a) 


Note  that  r2  =  —  and  r3  =  —  and  also  that  pava  = 

Pa  Pa 

p2v2,  etc.,  assuming  complete  intercooling. 

Laborious  precision  in  computing  the  work  done  on  or  by  com- 
pressed air  is  useless,  for  there  are  many  uncertain  and  changing 
factors;  n  is  always  uncertain  and  changes  with  the  amount  and 
temperature  of  the  jacket  water,  the  volumetric  efficiency,  or 
actual  amount  of  air  compressed,  is  usually  unknown,  the  value 
of  pa  varies  with  the  altitude,  and  r  is  dependent  on  pa. 

Art.  14.  Work  under  Variable  Intake  Pressure.  —  There  are 
some  cases  where  air  compressors  operate  on  air  in  which  the  in- 


28  COMPRESSED  AIR 

take  pressure  varies  and  the  delivery  pressure  is  constant.  This 
is  true  in  case  of  exhaust  pumps  taking  air  out  of  some  closed 
vessels  and  delivering  it  into  the  atmosphere.  It  is  also  the  con- 
dition in  the  "  return-air"  pumping  system  in  which  one  charge 
of  air  is  alternately  forced  into  a  tank  to  drive  the  water  out  and 
then  exhausted  from  the  tank  to  admit  water.  For  full  mathe- 
matical discussion  of  this  pump  see  Trans.  Am.  Soc.  C.  E.,  vol. 
54,  p.  19.  The  formulas  of  Arts.  14  and  15  were  first  worked 
out  to  apply  to  that  pumping  system. 

In  such  cases  it  is  necessary  to  determine  the  maximum  rate  of 
work  in  order  to  design  the  motive  power. 

First  assume  the  operation  as  being  isothermal.  Then  in  Eq. 
(1),  viz, 

W  = 


P 


px  is  variable,  while  v  and  pi  are  constant.  In  this  formula  W 
becomes  zero  when  ps  is  zero  and  again  when  px  —  pi,  since  log 
1  is  zero.  To  find  when  the  work  is  maximum,  differentiate  and 
equate  to  zero;  thus  differential  of 

Pi'—  Px  loge  Px)  =  V  j^loge  Pldpx -  (px  -p  +  log,  pxdpx.J  J  - 

Equate  this  to  zero  and  get  loge  pi  =  1  +  loge  px, 
or 

loge  —  =  1,  therefore  —  =  e  =  2.72. 

Px  Px 

That  is,  when  r  =  2.72  the  work  is  a  maximum. 

When  the  temperature  exponent  n  is  to  be  considered  the  study 
must  be  made  in  Eq.  (8),  viz. 

(8) 


Differentiating  this  with  respect  to  px  and  equating  to  zero, 

n-l 


the  condition  for  maximum  work  becomes  ( ^f )   7      =  n.     Insert 

\pL ' 

this  in  (8)  and  the  reduced  formula  becomes 

TT7  P*V 

W  =  npxv.  =  — -z — 


FORMULAS  FOR  WORK  29 

From  the  above  expression  for  maximum  the  following  results: 

When  n  =  1.41  the  maximum  occurs  when  r  =  3.26. 

When  n  =  1.25  the  maximum  occurs  when  r  =  3.05. 

When  n  =  1.00  the  maximum  occurs  when  r  =  2.72. 
In  practice  r  =  3  will  be  a  safe  and  convenient  rule. 

Exercise  14a.  —  Air  is  being  exhausted  out  of  a  tank  by  an  ex- 
haust pump  with  capacity  =  1  cu.  ft.  per  stroke.  At  the  begin- 
ning the  pressure  in  the  tank  is  that  of  the  atmosphere  =  14.7 
Ib.  per  sq'.  in.  Assume  the  pressure  to  drop  by  intervals  of  1  Ib. 
and  plot  the  curve  of  work  with  px  as  the  horizontal  ordinate  and 
W  as  the  vertical,  using  the  formula 

W  =  p.wlog.22- 

Px 

Exercise  146.  —  As  in  14a  plot  the  curve  by  Eq.  (8)  with  n  = 
1-25- 

Art.  15.  Exhaust  Pumps.  —  In  designing  exhaust  pumps  the 
following  problems  may  arise. 

Given  a  closed  tank  and  pipe  system  of  volume  V  under  pres- 
sure po  and  an  exhaust  pump  of  stroke  volume  v,  how  many 
strokes  will  be  necessary  to  bring  the  pressure  down  to  pm? 

The  analytic  solution  is  as  follows,  assuming  isothermal  condi- 
tions in  the  volume  V. 

The  initial  product  of  pressure  by  volume  is  pQV.  After  the 
first  stroke  of  the  exhaust  pump  this  air  has  expanded  into  the 
cylinder  of  the  pump  and  pressure  has  dropped  to  pi.  Under  the 
law  that  pressure  by  volume  is  constant; 


at  end  of  first  stroke, 


at  end  of  second  stroke, 

(V  +  v)p3  =  p2V,  or 

at  end  of  third  stroke,  etc. 
Finally 


Pm    = 


30  COMPRESSED  AIR 

This  is  inconvenient  for  solution  on  account  of  the  minus  charac- 
teristics.    Hence  it  is  better  to  write  it  thus: 

_       log  pm  -  log  p0 


log  V  -  log  (V  +  v) 
Now  change  sign  of  both  numerator  and  denominator  and  we  get 

log  po  -  log  pm 
log  (7  +  v)  -  log  V 

Example  15a. —  A  closed  tank  containing  100  cu.  ft.  of  air  at 
atmospheric  pressure  (14.7  Ib.)  is  to  be  exhausted  down  to  5  Ib. 
by  a  pump  making  1  cu.  ft.  per  stroke.  How  many  strokes  are 
required? 

Solution.— p0  =  14.7,  pm  =  5,  F  +  v  =  101  and  V  =  100. 

log  14.7  =  1.16136  log  101  =  2.00432 

log  5       =  0.69897;  log  100  =  2.00000 

0.46239  0.00432 

46239 


The  results  found  under  Arts.  14  and  15  serve  well  to  illustrate 
the  curious  mathematical  gymnastics  that  compressed  air  is  sub- 
ject to,  and  should  encourage  the  investigator  who  likes  such 
work,  and  should  put  the  designer  on  guard. 

Art.  16.  Efficiency  when  Air  is  Used  without  Expansion. — In 
many  applications  of  compressed  air  convenience  and  reliability 
are  the  prime  requisites,  so  that  power  efficiency  receives  little 
attention  at  the  place  of  application.  This  is  so  with  such  appara- 
tus as  rock  drills,  pneumatic  hammers  air  hoists  and  the  like. 
The  economy  of  such  devices  is  so  great  in  replacing  human  labor 
that  the  cost  in  power  is  little  thought  of.  Further,  the  necessity 
of  simplicity  and  portability  in  such  apparatus  would  bar  the 
complications  needed  to  use  the  air  expansively.  There  are  other 
cases,  however,  notably  in  pumping  engines  and  devices  of  vari- 
ous kinds,  where  the  plant  is  fixed,  the  consumption  of  air  con- 
siderable and  the  work  continuous,  where  neglect  to  work  the  air 
expansively  may  not  be  justified. 

In  any  case  the  designer  or  purchaser  of  a  compressed-air  plant 
should  know  what  is  the  sacrifice  for  simplicity  or  low  first  cost 
when  the  proposition  is  to  use  the  air  at  full  pressure  throughout 
the  stroke  and  then  exhaust  the  cylinder  full  of  compressed  air. 


FORMULAS  FOR  WORK  31 

Let  p  be  the  absolute  pressure  on  the  driving  side  of  the  piston 
and  pa  be  that  of  the  atmosphere  on  the  side  next  the  exhaust. 
Then  the  effective  pressure  is  p  —  pa  and  the  effective  work  is 
(p  —  Pa)  v,  while  the  least  possible  work  required  to  compress  this 
air  is  pv  loge  r. 

Hence  the  efficiency  is 

E    =    (P  ~  Pa)  E 
pV\Oger 

Dividing  numerator  and  denominator  by  pav  this  reduces  to 

E  =  ^f- ^  (18) 

r  log,  r 

This  is  the  absolute  limit  to  the  efficiency  when  air  is  used  without 
expansion  and  without  reheating.  It  cannot  be  reached  in 
practice. 

Table  VI  represents  this  formula.  Note  that  the  efficiency  de- 
creases as  r  increases.  Hence  it  may  be  justifiable  to  use  low- 
pressure  air  without  expansion  when  it  would  not  be  if  the  air 
must  be  used  at  high  pressure. 

Clearance  in  a  machine  of  this  kind  is  just  that  much  com- 
pressed air  wasted.  If  clearance  be  considered,  Eq.  (18)  becomes 

E  =  n    , r  ~  \  -  (I8o) 

(1  +  c)  r  loge  r 

where  c  is  the  percentage  of  clearance.  In  some  machines,  if 
this  loss  were  a  visible  leak,  it  would  not  be  tolerated. 

Art.  17.  Variation  of  Atmospheric  Pressure  with  Altitude.— 
In  most  of  the  formulas  relating  to  compressed-air  operations  the 
pressure  pa,  or  weight  waj  of  free  air  is  a  factor.  This  factor  varies 
slightly  at  any  fixed  place,  as  indicated  by  barometer  readings, 
and  it  varies  materially  with  varying  elevations. 

To  be  precise  in  computations  of  work  or  of  weights  or  volumes 
of  air  moved,  the  factors  pa  and  wa  should  be  determined  for  each 
experiment  or  test,  but  such  precision  is  seldom  warranted  further 
than  to  get  the  value  of  pa  for  the  particular  locality  for  ordinary 
atmospheric  conditions.  This,  however,  should  always  be  done. 
It  is  a  simple  matter  and  does  not  increase  the  labor  of  computa- 
tion. In  many  plants  in  the  elevated  region  pa  may  be  less  than 
14.0  Ib.  per  square  inch,  and  to  assume  it  14.7  would  involve  an 
error  of  more  than  5  per  cent. 

Direct  reading  of  a  barometer  is  the  easiest  and  usual  way  of 


32  COMPRESSED  AIR 

getting  atmospheric  pressure,  but  barometers  of  the  aneroid  class 
should  be  used  with  caution.  Some  are  found  quite  reliable,  but 
others  are  not.  In  any  case  they  should  be  checked  by  compari- 
son with  a  mercurial  barometer  as  frequently  as  possible. 

If  m  be  the  barometer  reading  in  inches  of  mercury  and  F  be 
the  temperature  (Fahrenheit),  the  pressure  in  pounds  per  square 
inch  is 

pa  =  0.4912  m[l  -  0.0001  (F  -  32)]  (19) 

NOTE.  —  One  cubic  inch  of  mercury  at  32°F.  weighs  0.4912  Ib. 
The  information  in  Table  II  will  usually  obviate  the  need  of 
using  Eq.  (19). 

In  case  the  elevation  is  known  and  no  barometer  available  the 
problem  can  be  solved  as  follows  : 
Let  ps  =  pressure  of  air  at  sea  level, 
ws  =  weight  of  air  at  sea  level, 
Px,  MX  be  like  quantities  for  any  other  elevation. 

Then  in  any  vertical  prism  of  unit  area  and  height  dh  we  have 

dpx  =  wxdh. 
But 

Wx         Px     ,->          f  j  Ws        ,, 

-  =  —  ;  therefore  dpx  =  —pxdh, 
wa       pa'  ps 

or 

dh  =  P^^  and  therefrom  h  =  ^  X  log  ^, 

Ws    px'  Ws  &pa 

where  pa  is  the  pressure  at  elevation  h  above  seal  level.  Sub- 
stitute for  ws  its  equivalent 


e  get 
Whence 


Making  ps  =  14.745  and  adopting  to  common  logarithm   and 
Fahrenheit  temperatures, 

Iogl,  pa  =  1.16866  -  —--  (20) 

Table  V  is  made  up  by  formula  (20). 


CHAPTER  II 
MEASUREMENT  OF  AIR 

Art.  18.  General  Discussion. — Progress  in  the  science  of  com- 
pressed-air production  and  application  has  evidently  been  hin- 
dered by  a  lack  of  accurate  data  as  to  the  amount  of  compressed 
air  produced  and  used. 

The  custom  has  been  almost  universal  of  basing  computations 
on,  and  of  recording  results  as  based  on,  catalog  rating  of  compres- 
sor volumes — that  is,  on  piston  displacement. 

The  evil  would  not  be  so  great  if  all  compressors  had  about  the 
same  volumetric  efficiency,  but  it  is  a  fact  that  the  volumetric 
efficiency  varies  from  60  to  90  per  cent.,  depending  on  the  make, 
size,  condition  and  speed  of  the  machine,  no  wonder,  then,  that 
calculations  often  go  wrong  and  results  seem  to  be  inconsistent. 

There  are  problems  in  compressed-air  transmission  and  use 
for  the  solution  of  which  accurate  knowledge  of  the  volume  or 
weight  of  air  passing  is  absolutely  necessary.  Chief  among  these 
are  the  determination  of  friction  factors  in  air  pipes  and  the 
efficiency  of  compressors,  pumps,  air  lifts,  fans,  etc. 

Purchasers  may  be  imposed  upon,  and  no  doubt  often  are, 
in  the  purchase  of  compressors  with  abnormally  low  volumetric 
efficiencies.  Contracts  for  important  air-compressor  installation 
should  set  a  minimum  limit  for  the  volumetric  efficiency,  and  the 
ordinary  mechanical  engineer  should  have  knowledge  and  means 
sufficient  to  test  the  plant  when  installed. 

There  is  little  difficulty  in  the  measurement  of  air.  The 
calculations  are  a  little  more  technical,  but  the  apparatus  is  as 
simple  and  the  work  much  less  disagreeable  than  in  measurements 
of  water. 

At  this  date  (1917)  practice  does  not  seem  to  have  settled  on 
a  standard  method  of  measuring  quantities  of  air;  but  current 
literature  shows  that  the  subject  is  receiving  what  seems  to  be 
the  long-delayed  attention  that  it  deserves. 

In  any  case  where  the  air  or  gas  to  be  measured  will  have  a  con- 
stant density  and  it  is  necessary  only  to  get  the  rate  of  flow  at  any 
time,  the  apparatus  and  methods  applicable  would  be  as  simple 
3  33 


34 


COMPRESSED  AIR 


as  those  applied  in  measuring  water,  but  the  problem  is  not  so 
simple  when  it  is  necessary  to  record  the  total  flow  (weight)  dur- 
ing a  considerable  time  during  which  the  pressure  and  density 
may  vary  between  wide  limits.  Though  there  are  some  appara- 
tus that  the  makers  claim  will  do  this,  the  problem  does  not  seem 
to  have  been  solved  in  a  satisfactory  way. 

Art.  19.  Apparatus  for  Measuring  Air. — Several  methods  of 
measuring  the  rate  of  flow  of  air  at  the  time  of  observation  (or 
with  pressure  and  temperature  constant),  that  have  been  pro- 
posed and  tried,  will  be  briefly  noted  as  follows:1 

(a)  The  Venturi  Meter. — The  principle  is  identical  with  that  of 
the  venturi  water  meter,  but  it  is  necessary  to  determine  the 
coefficient  over  a  range  covering  all  pressures  under  which  it  may 
be  used.  This  coefficient  may  not  change  with  pressure,  but  if 
so  the  fact  has  not  been  ascertained. 

(6)  The  ''Swinging  Gate," 
Fig.  8a. — The  air  flowing  in 
the  direction  of  the  arrow 
swings  the  gate  open.  The 
angle  of  opening  depends  on 
the  weight  of  the  gate,  and 
on  the  density  and  velocity 
of  the  air.  Every  gate  will 
have  a  special  set  of  coeffi- 
cients and  these  would  have  to  cover  the  whole  field  of  velocities 
and  densities. 

(c)  The  Thermal  Method. — In  this  scheme  the  air  is  passed 
through  an  enlargement  of  the  pipe  in  which  there  is  placed  an 
exposure  of  a  great  surface  of  wire,  the  wire  being  heated  by  a 
measured  electric  current.     The  temperature  of  the  air  is  meas- 
ured before  and  after  passing  over  the  heated  wire.     The  weight  of 
air  passing  can  be  expressed  in  terms  of  the  rise  of  temperature 
and  the  electric  current  absorbed.     The  objections  are:  Expen- 
sive apparatus,  requiring  great  sensitiveness,  and  liability  to  error 
through  various  sources,  among  which  is  the  humidity  of  the  air. 

(d)  Mechanical    Meters. — This    class    includes    common    gas 
meters.     They  are  satisfactory  for  commercial  purposes  and  for 
such  capacities  as  are  covered  by  stock  sizes.     For  large  volumes 
they  become  expensive  and  the  coefficient  is  always  liable  to 
variation,  that  is,  the  record  may  become  inaccurate  due  to 

1  See  Compressed  Air  Magazine,  vol.  16,  p.  6255. 


FIG.  8a. 


MEASUREMENT  OF  AIR  35 

corrosion  or  fouling  of  the  mechanisms.  Such  meters  show  only 
the  total  volume  that  has  passed  between  readings  but  unless  the 
pressure  and  temperature  are  constant  the  record  does  not  show 
the  quantity  or  weight. 

As  stated  above,  none  of  these  methods  will  apply  when  it  is 
necessary  to  determine  the  total  weight  passing  during  a  pro- 
longed time  in  which  the  pressure  varies.  If  in  cases  (a)  and 
(6)  the  pressure  is  constant  and  the  velocity  only  changes,  a 
continuous  recording  apparatus  could  be  attached  to  make  a 
graph  giving  time  and  differential  head  in  case  (a)  or  time  and 
swing  of  gate  in  case  (6)  from  which  cards  the  total  volume  could 
be  integrated.  If  simultaneously  another  graph  be  taken  show- 
ing time  and  pressure  the  two  could  be  used  to  work  out  weights. 

If  inventors  could  go  this  far,  they  could  afford  to  neglect 
temperatures  in  commercial  work.  However,  the  cost  of  the 
apparatus  and  the  labor  of  determining  the  proper  coefficients 
seem  to  bar  any  of  the  above  from  general  use. 

Art.  20.  Measurement  by  Standard  Orifices. — For  reasons  of 
economy,  simplicity  and  accuracy,  it  seems  that  practice  will 
settle  on  the  standard  orifice  for  determining  the  flow  of  air. 
For  this  reason  the  method  and  apparatus  are  described  in 
detail. 

The  standard  orifice  is  the  same  as  that  specified  for  the  meas- 
urement of  water,  that  is,  an  orifice  in  a  thin  plate  (or  with  sharp 
edges).  In  this  article  only  circular  orifices  will  be  considered. 
These  may  be  cut  in  any  sheet  metal  up  to  J£  in.  thick.  The 
standard  conditions  shall  be  that  the  drop  in  pressure  in  passing 
through  the  orifice  shall  not  exceed  6  in.  head  of  water. 

With  this  restriction  of  conditions  the  change  of  temperature 
and  of  density  of  the  air  while  passing  the  orifice  may  be  neglected 
in  commercial  operations  without  appreciable  error.  This 
very  much  simplifies  the  formulas  and  reduces  the  chances  of 
error. 

With  these  standards,  experiments  show  coefficients  for  air 
more  nearly  constant  than  for  water. 

Art.  21.  Formula.  Standard  Orifice  under  Standard  Condi- 
tions.— 

Let  p  =  absolute  pressure  of  air  approaching  the  orifice  =  rpa, 
Q  =  weight  of  air  passing  per  second, 
w  =  weight  of  a  cubic  foot  of  air  at  pressure  p, 
d  =  diameter  of  orifice  in  inches, 


36  COMPRESSED  AIR 

i  =  pressure  as  read  on  water  gage  in  inches, 
t  =  absolute  temperature  of  air  (F), 
c  —  experimental  coefficient. 

When  change  of  temperature  and  of  density  can  be  neglected, 
the  theoretic  velocity  through  an  orifice  is 


where  h  is  the  head  of  air  of  uniform  density  (w)  that  would 

produce  the  pressure  head  i. 

Hence 

i  62.5    ,,       ,  L     i   6.25 

h  =        therefore  s  =: 


But   Q  =  w  X  a  X  s  where  a  equals  the  area  of   orifice  in 

d* 
square  feet  =  TT  4^-144'     Inserting  these  values  and  putting  w 

under  the  radical,  there  results 


but 

W  =  53M' 
therefore 


Q  =  0.0136d2<\  /-  rpa      where  pa'  is  in  pounds  per  square  foot, 


=  0.1639d2A/-  rpa     where  pa  is  in  pounds  per  square  inch. 

To  this  must  be  applied  the  experimental  coefficient  c  so  the 
formula  becomes 

Q  =  c  X  0.1639d2^|j  rpa  (21) 

For  distilled  water  and  dry  air  the  equation  would  be 


Q  =  c  X  0.1645d2A/-  rpa. 

In  very  precise  determinations  the  weight  of  air  should  be 
determined  to  accord  with  its  humidity  (see  Art.  8a).  This 
value  of  w  would  then  go  into  Eq.  (a)  above. 

When  working  with  an  orifice  set  in  a  low-pressure  drum,  the 


MEASUREMENT  OF  AIR 


37 


product  rpa  can  be  most  readily  gotten  by  adding  to  pa  the 
quantity  0.036i  which  is  the  pressure  on  a  square  inch  due  to 
a  head  i.  Thus  rpa  =  pa  +  0.036^. 

If  mercury  be  the  liquid  in  the  U-gage  and  barometer  heights 
be  inches  of  mercury,  then 

Q  =  c  X *A 


where  h  =  barometer  height  +  i  (i  being  inches  of  mercury). 

It  will  often  be  convenient  to  compute  the  weight  of  air  when 
pressure  is  in  inches  of  mercury. 
Then 


=  1.321 


(21o) 


The  apparatus  to  be  used  in  combination  with  this  formula 
depends  on  whether  the  measured  air  is  to  be  discharged  directly 
into  the  free  atmosphere  or  is  to  be  retained  in  the  pipe  system 
under  pressure. 

Art.  22.  Apparatus  for  Measuring  Air  at  Atmospheric  Pres- 
sure.— This  is  the  simpler  of  the  two  cases  and  is  the  one  most 
easily  applied  in  a  single  test  of  an  air  compressor.  The  essentials 
are  indicated  in  Fig.  9. 


j/B 

N^ 

8 

;  1- 

B 

b 

b 

b 

"3 

FIG.  9. 

A  =  compressed-air  pipe, 
B  =  closed  box  or  cylinder, 
T  =  throttle, 

b  =  baffle  boards  or  screen, 
H  =  thermometer, 
C  =  cork, 

0  =  orifice  in  thin  metal  plate  (Standard), 
U  =  bent  glass  tube  containing  colored  water, 
G  =  scale  of  inches. 


38  COMPRESSED  AIR 

The  box  B  may  be  made  of  any  light  material,  wood  or  metal. 
The  pressure  will  be  only  a  few  ounces  and  the  tendency  to  leak 
correspondingly  slight.  The  purpose  of  the  throttle  T  is  to 
control  the  pressure  against  which  the  compressor  works.  The 
appropriate  orifice  can  be  determined  by  a  preliminary  compu- 
tation, assuming  i  at  say  3  in.,  or  use  Plate  I. 

Art.  23.  Coefficients  for  Large  Orifices. — Experiments  were 
made  at  Missouri  School  of  Mines  in  1915  to  determine  the  co- 
efficient, c,  to  apply  in  formula  (21)  in  case  of  large  orifices  up  to 
30  in.  in  diameter  and  30  by  30  in.  square.  The  scheme  being 
as  follows:1 

Having  a  fan  or  blower  of  capacity  and  pressure  sufficient  for 
the  purpose,  direct  the  discharge  into  a  conduit  across  which 
place  one  partition  containing  the  appropriate  number  of  small 
standard  orifices  for  which  the  coefficient  is  known  and  in  an- 
other partition  place  the  large  orifice.  Then  the  same  quantity 
of  air  passes  through  the  group  of  small  orifices  and  the  single 
large  orifice,  and  by  observing  the  water  gage  at  each  partition 
the  relation  between  the  coefficients  can  be  found  thus : 

Let  GI  be  the  unknown  coefficient  of  the  large  orifice, 
C2  be  the  known  coefficient  of  the  small  orifices, 
n  be  the  number  of  small  orifices  open, 
d  be  the  diameter  of  the  small  orifices, 
D  be  the  diameter  of  the  large  orifices. 

Then  by  formula  (21) 

Q  =  ci  X  0.1639D2A/*1  pi  =  c2  X  0.1639nd 

\  t\ 


Sub  1  and  sub  2  indicating  symbols  at  the  large  and  small  orifice 
partitions,  respectively. 

Now  it  can  be  shown  that  where  the  drop  in  pressure  is  only 
a  few  inches  (water  gage)  the  factors 


may  be  taken  as  equal,  especially  so  if  the  water  gages  be 
nearly  equal  at  the  two  partitions.  Hence  we  may  express 
the  relation  of  the  two  coefficients,  thus 

.  „  ,'nd* 
(_/ i  — 


Missouri  School  of  Mines  Bulletin,  vol.  2,  No.  2,  November,  1915. 


MEASUREMENT  OF  AIR 


39 


40  COMPRESSED  AIR 

Similarly,  when  the  large  orifice  is  rectangular  with  area  =  a, 

nird2 
i~ 

For  convenience  let  K  represent  the  factor  in  parenthesis;  then 
Ci  =  KC%. 

In  the  experiments  referred  to,  the  following  results  were 
obtained : 

Seventy-seven  3^-in.  orifices  passing  to  one  30-in.  round K  =  1.01. 

Fifty  3^-in.  orifices  passing  to  one  24-in.  round K  =  1 .00. 

Twenty-six  33^-in.  orifices  passing  to  one  18-in.  round K  =  0.996. 

Fifty-eight  3^-in.  orifices  passing  to  one  18  by  30-in.  rectangle .  K  =  1 . 005. 

Sixty  3^-in.  orifices  passing  to  one  24  by  24-in.  rectangle K  =  1.014. 

Thirty-four  3>£-m.  orifices  passing  to  one  18  by  18-in.  rectangle .  K  =  0 . 998. 

From  the  above  it  is  evident  that  for  commercial  purposes  the 
coefficients  for  these  large  orifices  may  be  taken  as  equal  that  of  a 
3^-in.  orifice  (see  Table  VIII).  Errors  in  reading  water  gages 
will  probably  exceed  that  made  by  such  an  assumption. 

Accepting  the  coefficients  shown  in  Table  VIII,  those  for  large 
orifices  are  as  shown  in  Table  Villa. 

As  a  result  of  these  experiments  it  is  evident  that  large  orifices, 
conforming  to  standard  conditions,  can  be  used  with  as  much 
accuracy  as  in  case  of  small  ones. 

This  being  accepted,  there  is  available  for  testing  large  fans 
and  blowers  the  most  reliable  of  all  methods  of  measuring  the 
flow  of  fluids,  that  is  orifice  measurement.  Note  that  one  30-in. 
round  orifice  will  pass  about  25,000  cu.  ft.  per  minute  under 
4-in.  water  pressure. 

Where  very  large  fans  are  to  be  tested  several  orifices  can  be 
set  in  a  conduit  wall.  For  such  cases  accurately  constructed 
wood  orifices  would  probably  be  entirely  reliable  and  could  be 
put  in  at  moderate  cost. 

Art.  23a.  Notes  on  Water  Gages. — Experience  with  water 
gages,  and  in  efforts  to  improve  on  the  plain  water  gage,  while 
doing  this  work  may  be  of  interest. 

In  such  a  gage  (any  liquid)  when  oscillations  (not  gradual 
changes  of  pressure)  interfere  with  the  readings,  a  few  bird  shot 
(filling  the  tube  about  an  inch)  will  prevent  oscillations  and  yet 
permit  sufficient  sensitiveness  under  changing  pressure. 

Any  coloring  matter  is  liable  to  cause  error  by  changing  the 
specific  gravity  of  the  water. 


MEASUREMENT  OF  AIR  41 

Makers  of  some  special  gages  recommend  the  use  of  gasoline 
of  known  specific  gravity,  instead  of  water,  as  it  is  lighter  and 
therefore  more  sensitive.  On  trial  it  was  found  that  if  the  two 
columns  of  the  gage,  above  the  liquid,  are  unequal  in  height,  the 
presence  of  gasoline  gas  in  the  high  column  will  unbalance  the 
fluid  columns  and  cause  error.  Often  one  arm  of  the  gage  is 
continued  in  a  rubber  tube.  This  will  in  effect  be  an  extension 
of  the  column.  In  a  gage  in  which  the  two  columns  have  equal 
bore,  or  caliber,  throughout,  the  sum  of  the  two  column  readings 
will  be  constant  as  long  as  the  volume  of  liquid  in  the  gage  does 
not  change.  In  attempting  to  utilize  this  fact  in  a  gage  filled 
with  gasoline  it  was  found  that  the  gasoline  evaporated  so  fast 
as  to  render  the  scheme  inapplicable.  The  same  liability  to 
inaccuracies  exist  in  any  of  the  combination  gages  in  which  both 
water  and  gasoline  are  used. 

Where  much  work  is  to  be  done  while  pressures  are  changing, 
the  best  scheme  is  to  get  a  gage  in  which  the  sum  of  the  readings 
is  constant;  use  water  or  mercury;  find  the  sum  of  the  two  column 
readings  and  then  read  only  one  column. 

Let  s  =  sum  of  column  reading, 

h  =  reading  of  upper  column  of  liquid, 
I  =  reading  of  lower  column  of  liquid. 
Then  i  =  2  (h  -  lAs)  or  i  =  2  (^s  -  I). 

Experience  in  this  work  in  which  thousands  of  readings  of 
fluid  pressure  gages  have  been  made  under  a  variety  of  conditions 
and  with  a  variety  of  gages,  leads  those  who  have  done  most  of 
the  work  to  the  conclusion  that  most  reliable  results  can  be  got- 
ten with  pure  water  in  a  plain  U-tube  fastened  vertically  over 
a  scale  tacked  to  a  plane  board;  the  arms  of  the  tube  about  2-in. 
apart  and  the  horizontal  ruling  of  the  scale  extending  under  both 
arms  of  the  gage.  The  readings  to  be  taken  with  the  assistance 
of  a  small  draftsman's  triangle  held  with  the  side  resting  against 
the  vertical  glass  tube  and  edge  against  the  scale,  parallax  being 
avoided  by  bringing  the  eye  so  that  the  upper  edge  of  the  tri- 
angle and  the  lines  on  the  scale  are  projected  parallel  and  both 
seen  crossing  the  gage  column  as  illustrated  in  the  photograph. 
(Note  that  the  eye  of  the  camera  was  not  in  the  correct  position.) 

Art  24.  Apparatus  for  Measuring  Air  Under  Pressure  with 
Standard  Orifices. — In  the  ordinary  case  when  it  is  desired  to 
know  the  quantity  of  compressed  air  passing  through  a  pipe  with- 


42 


COMPRESSED  AIR 


out  sacrificing  the  pressure,  the  orifice  drum  must  be  made  strong 
enough  to  withstand  the  high  pressure  and  the  U-gage  de- 
scribed in  the  previous  case  must  be  replaced  by  a  differential 
gage  which  must  also  be  strong  enough  to  withstand  the  pres- 
sure. The  essentials  are  embodied  in  the  illustration,  Fig.  10, 


FIG.  9a. — Method  of  reading  water  gages. 

which  also  suggests  a  convenient  scheme  for  attachment  to  an 
air  main. 

The  several  essentials  are: 

ViVzVz  =  valves  for  controlling  the  path  of  the  air, 
U  =  unions  for  detaching  apparatus, 

—  baffles  for  steadying  the  current  of  air, 


MEASUREMENT  OF  AIR 


43 


0  =  orifice, 

T  =  thermometer  set  through  a  gland, 
G  =  pressure  gage, 

gg2  =  glass  columns  of  the  differential  gage, 
C  =  cocks  for  convenience  in  manipulating  the  differ- 
ential gage. 

The  manipulation  of  the  apparatus  Fig.  10,  is  as  follows: 
To  charge  the  differential  gage  close  Ci,  C4  and  C5,  open  <72  and 
C3  and  pour  in  the  desired  amount  of  liquid.     Then  close  C2  and 
Cz  and  open  C*  and  Cs. 

To  pass  the  air  through  the  measuring  drum,  open  V2  and  73 
and  close  V\. 


NOTE:  Both  legs  of  the  gage 
should  be  tapt  into  the  drum  close 
beside  the  orifice. 


FlG. 


Art.  25.  Coefficients  and  Orifice  Diameters  for  Measurements 
at  High  Pressures. — Unless  evidence  to  the  contrary  is  shown,  it 
is  reasonable  to  assume  that  the  same  coefficients  would  apply  to 
the  orifice  in  the  high-pressure  drum,  Fig.  10,  that  have  been 
determined  for  the  low-pressure  drum,  Fig.  9.  However,  for  the 
same  Q,  i,  t  and  c  the  diameters,  d,  must  differ  according  to  the 
following : 


44  COMPRESSED  AIR 

Let  di  and  pi  be  the  orifice  diameter  and  air  pressure  respect- 
ively in  the  high-pressure  drum,  and  note  that  the  pressure  in 
the  low-pressure  drum  may  be  taken  for  this  purpose  as  pa. 
Then 

Qi  =  Q  =  C  X  0.1639rf2  Jj  pa  =  c  X  0.1639dis 

\  * 

Whence     . 

*  =  &  (22) 

since  pi/pa  =  r. 

By  this  relation  the  appropriate  orifice  can  be  determined  from 
the  curve,  Plate  I,  by  dividing  the  diameter  ordinate  by  (r)w. 

The  size  drum  necessary  to  measure  a  given  volume  of  free  air 
when  under  pressure  is  not  as  large  as  might  be  supposed  before 
computations  are  made.  For  instance,  with  i  =  3  in.,  T  =  60°F. 
and  c  =  0.60,  a  3-in.  orifice  will  pass  570  cu.  ft.  of  free  air  per 
minute  when  compressed  to  100  Ib.  If  this  3-in.  orifice  be  placed 
in  a  drum  8  in.  in  diameter,  the  velocity  of  the  compressed  air 
within  the  drum  will  be  3.5  ft.  per  second,  which  is  conservative. 

Example  25. — In  a  run  with  the  apparatus  shown  in  Fig.  9,  the 
following  were  the  records:  d  =  2.32  in.,  i  =  4.6  in.,  T  =  186°F. 
inside  drum,  T  =  86°F.  in  free  air,  elevation  =  1,200  ft. 

Find  the  weight  and  volume  of  air  passing  per  minute. 

Solution.— From  Table  II  interpolating  for  86°  in  the  line 
with  1,200  elevation  we  get  wa  =  0.0700  and  pa  =  14.1.     Add  to 
pa  the  pressure  due  to  i  (  =  0.036  X  4.6)  and  we  get  pa  =  14.26. 
In  Table  VIII  the  coefficient  for  d  =  2.32  and  i  =  4.6  is  0.599. 
These  numbers  inserted  in  Eq.  (21)  give 

Q  =  0.599  X  0.1639  X  (2.32)2^^r  X  14.26  =  0.1684  Ib.  per 

,  0.1684  X  60 
second;  and  -      ~  „- —  =  144.3  cu.  ft.  per  minute  of  free  air. 

Should  there  be  doubt  about  the  coefficients  being  the  same  for 
both  high-  and  low-pressure  drums,  and  we  are  willing  to  accept 
these  now  published  for  low-pressure  drums,  we  can  determine 
that  of  the  high-pressure  drum  by  placing  the  two  drums  in 
tandem,  the  same  quantity  of  air  passing  through  the  high-  and 
low-pressure  drums  in  succession.  Then  letting  sub  1  refer  to 
the  high-pressure  drum  we  have  the  equation, 

Q  =  ci  X  0.1639di2^  pi  =  c  X  0.1639d2  ^-pa. 


MEASUREMENT  OF  AIR  45 

Whence 

Cl2  =  c2(!)4J>^  (23) 

\di/    ^l  t  pi 

In  extensive  experiments  at  Missouri  School  of  Mines  in  1915, 
the  coefficients  proved  to  be  equal  so  far  as  practical  applications 
would  be  concerned  though  the  high-pressure  coefficients  seemed 
to  be  slightly  less.  The  experiments  were  not  conclusive.  See 
description  of  oil  differential  gage,  Appendix  D. 

In  advocating  the  standard-orifice  method  of  measuring  air  it 
should  be  noted  that  the  coefficient  of  an  orifice  is  not  liable  to 
change  with  time  and  that  the  necessary  apparatus  can  be  made 
up  in  any  reasonably  well-equipped  shop  of  a  compressed-air 
plant. 

The  method  as  presented  is  adapted  only  to  show  the  rate  of 
flow  at  the  time  of  observation.  To  determine  the  quantity 
passed  during  any  prolonged  period  a  continuous  recording 
apparatus  would  have  to  be  attached  that  would  show  both  the 
value  of  i  and  of  p.  The  factor  t  might  be  assumed  constant  in 
most  cases  in  practice  but  even  then  the  apparatus  would  be 
intricate,  delicate  and  expensive. 

It  may  be  stated  then  that  there  are  no  satisfactory  means  now 
available  to  measure  the  quantity  of  air  passed  during  a  definite 
time  where  pressure  and  velocities  vary.  However,  the  obstacles 
are  not  insurmountable. 

Art.  26.  Discharge  of  Air  through  Orifice.  Considerable 
Drop  in  Pressure.  —  Referring  to  Figs.  9  and  10,  when  the  differ- 
ence in  pressures  pi  and  p%  is  considerable,  we  cannot  neglect  the 
change  of  density  and  of  temperature. 

To  analyze  this  case  we  must  start  from  the  equations  of  energy 
at  sections  1  and  2,  inside  and  outside  the  orifice,  the  energy  in 
each  case  being  part  kinetic  and  part  potential. 

Thus 

f,+  ^  =  f  +  ^  (a) 

or 


where  c  =  53.35  for  1  Ib.  (see  Art.  2). 
Whence 


46  COMPRESSED  AIR 

Now  in  any  practical  case  the  velocity  of  approach  Si  to  the 

orifice  can  be  made  so  small  that  the  numerical  value  of  ~-  is 

20 

so  small  as  compared  with  cti2  that  it  can  be  neglected,  if  desired, 

«22 

without  appreciable  error;  but  not  so  with  the  quantity  -^—  • 

*9 
Hence  we  may  write 


Substituting  for  tz  its  value  from  Eq.  (12a),  viz., 


we  get 

n-l 


where  rx  is  the  ratio  —  when  the  escape  is  into  free  air. 
Pi 

The  weight  passing  per  second  is  Q  =  waaSz  where  a  is  the 

/r\ 

area  of  orifice  and  wa  =  — ~  in  which  again  substitute  for  t%  its 

el 

value  as  above.     These  substitutions  give 


This  is  a  max.  when 

rx 


When  n  =  1.41i  Q  is  max.  when  rx  =  0.526. 

When  n  =  1.25i  Q  is  max.  when  rx  =  0.555. 

Any  such  law  as  this  could  not  have  been  suspected  except  by 
mathematical  analysis,  and  seems  contrary  to  what  would  other- 
wise have  been  supposed.  Yet  experiment  seems  to  show  that 
it  is  correct. 

Equation  (24)  is  not  recommended  as  a  formula  for  practical 
application  in  measuring  air. 

Art.  27.  Air  Measurement  in  Tanks. — The  amount  of  air 
put  into  or  taken  out  of  a  closed  tank  or  system  of  tanks  and 


MEASUREMENT  OF  AIR  47 

pipes,   of  known   volume,    can   be   accurately   determined   by 
Eq.  (3),  viz., 

PaVa   _ta  _   Pxta  Vx 

—      or  Va  —  ,  " 

PxVx          tx  pa      tx 

The  process  would  be  as  follows :  Determine  the  volumes  of  all 
tanks,  pipes,  etc.,  to  be  included  in  the  closed  system,  open  all 
to  free  air  and  observe  the  free-air  temperature;  then  switch 
the  delivery  from  the  compressor  into  the  closed  system;  count 
the  strokes  of  the  compressor  until  the  pressure  is  as  high  as 
desired;  then  shut  off  the  closed  tank  and  note  pressure  and  tem- 
peratures of  each  separate  part  of  the  volume.  Then  the  formula 
above  will  give  the  volume  of  free  air  which  compressed  and 
heated  would  occupy  the  tanks.  From  this  subtract  the  volume 
of  free  air  originally  in  the  tanks ;  the  remainder  will  be  what  the 
compressor  has  delivered  into  the  system.  Note  that  the  com- 
pressor should  be  running  hot  and  at  normal  speecl  and  pressure 
when  the  test  is  made  for  its  volumetric  efficiency. 

Usually  the  temperature  changes  will  be  considerable,  but  if 
the  system  is  tight,  time  can  be  given  for  the  temperature  to 
come  back  to  that  of  the  atmosphere,  thus  saving  the  necessity 
of  any  temperature  observations. 

Where  a  convenient  closed-tank  system  is  available,  this 
method  is  recommended. 

This  method — that  is,  Eq.  (3)  as  stated  above — was  used  to 
determine  the  quantity  of  air  passing  the  orifices  in  the  experi- 
ments by  which  the  coefficients  were  determined  as  given  in 
Art.  21,  Table  VIII. 

The  varying  volumetric  efficiencies  with  changes  of  tempera- 
ture and  pressures  can  be  shown  very  impressively  by  starting 
with  compressor  cool  and  the  air  in  tanks  at  atmospheric  pres- 
sure. Then  note  the  number  of  revolutions  that  bring  the  pres- 
sure up  to  say  20,  40,  60,  80  lb.,  and  so  get  the  data  for  volumetric 
efficiencies  in  each  interval.  In  the  first  it  may  be  found  as  high 
as  95  per  cent,  while  in  the  last  interval  it  may  fall  below  60  per 
cent,  in  small  compressors.  Of  course,  that  in  the  last  interval 
is  that  by  which  the  compressor  should  be  judged. 

Example  27. — A  tank  system  consists  of  one  receiver  3  ft.  in 
diameter  by  12  ft.,  one  air  pipe  6  in.  by  40  ft.,  one  4  in.  by  4,000 
ft.  and  a  second  receiver  at  end  of  pipe  2  ft.  in  diameter  by  8  ft. 
A  compressor  12  by  18  in.  with  1^-in.  piston  rod  puts  the  air 


48  COMPRESSED  AIR 

from  1,250  revolutions  into  the  system,  after  which  the  pressure 
is  80-gage  and  temperature  in  first  receiver  200°,  while  in  other 
parts  of  the  tank  system  it  is  60°.  Temperature  of  outside  air 
being  50°,  pa  =  14.5  per  square  inch.  Find  volumetric  efficiency 
of  the  compressor. 

Solutions. — Volumes  (from  Table  X) : 

First  receiver 84.84  cu.  ft. 

6-in.  pipe 7.841 

4-in  pipe 349.20    382.16 

Second  receiver. .  25 . 12  J 


Total  ...........  467.00  in  tank  system. 

Piston  displacement  in  one  revolution  =  2.338  cu.  ft.  (piston 
rod  deducted). 

By  formula  va  =  {—  -  }  X  —  note  that  the  quantity  in  paren- 

\  Pa  I  lx 

thesis  is  constant  and  therefore  a  slide  rule  can  be  conveniently 
used,  otherwise  work  by  logarithms 

.'.:-  .  (80  +  14.5)  (460  +  50)  w       84.84  ._  0 

v.  m  first  receiver  =  --  -  X  =  417.2 


va  in  6-in.  pipe,  4-in.  pipe  and  second  receiver  with  total 

volume  382.16  and  t  =  60°  =  .  2,447.1 


Total 2,864.3 

Original  volume  of  free  air s 467.0 


Volume  of  free  air  added. 2,397.3 


2,397.3  ^  2.338  =  1,028. 
Therefore  the  volunxetric  efficiency  is 

E  =  1,028  -f-  1,250  =  82  per  cent. 


CHAPTER  III 
FRICTION  IN  AIR  PIPES 

Art.  28. — In  the  literature  on  compressed  air  many  formulas 
can  be  found  that  are  intended  to  give  the  friction  in  air  pipes 
in  some  form.  Some  of  these  formulas  are,  by  evidence  on  their 
face^  unreliable,  as  for  instance  when  no  density  factor  appears; 
the  origin  of  others  cannot  be  traced  and  others  are  in  incon- 
venient form.  Tables  claiming  to  give  friction  loss  in  air  pipes 
are  conflicting,  and  reliable  experimental  data  relating  to  the 
subject  are  quite  limited. 

In  this  chapter  are  presented  the  derivation  of  rational  for- 
mulas for  friction  in  air  pipes  with  full  exposition  of  the  assump- 
tions on  which  they  are  based.  The  coefficients  were  gotten 
from  the  data  collected  in  Appendix  B. 

Art.  29.  The  Formula  for  Practice. — The  first  investigation  will 
be  based  on  the  assumption  that  volume,  density  and  tempera- 
ture remain  constant  throughout  the  pipe. 

Evidently  these  assumptions  are  never  correct;  for  any  de- 
crease in  pressure  is  accompanied  by  a  corresponding  increase 
in  volume  even  if  temperature  is  constant.  (The  assumption  of 
constant  temperature  is  always  permissible.)  However,  it  is 
believed  that  the  error  involved  in  these  assumptions  will  be 
less  than  other  unavoidable  inaccuracies  involved  in  such 
computations. 

Let  /  =  lost  pressure  in  pounds  per  square  inch, 
I  =  length  of  pipe  in  feet, 
d  =  diameter  of  pipe  in  inches, 
s  =  velocity  of  air  in  pipe  in  feet  per  second, 
r  =  ratio  of  compression  in  atmospheres, 
c  =  an  empirical  coefficient  including  all  constants. 

Experiments  have  proved  that  fluid  friction  varies  very  nearly 

with  the  square  of  the  velocity  and  directly  with  the  density. 

Hence  if  k  be  the  force  in  pounds  necessary  to  force  atmospheric 

air  (r  =  1)  over  1  sq.  ft.  of  surface  at  a  velocity  of  1  ft.  per 

4  49 


50  COMPRESSED  AIR 

second,  then  at  any  other  velocity  and  ratio  of  compression  the 
force  will  be 

F!  =  ks*r, 

and  the  force  necessary  to  force  the  air  over  the  whole  interior 
of  a  pipe  will  be 

F  =  ^l  X  fcr.', 

and  the  work  done  per  second,  being  force  multiplied  by  distance, 
is 

Work  =  ~  X  krs*. 

Now  if  the  pressure  at  entrance  to  the  pipe  is  /  Ib.  per  square 
inch  greater  than  at  the  other  end,  the  work  per  second  due  to 
this  difference  (neglecting  work  of  expansion  in  air)  is 

Work  =  /  —  8. 

Equating  these  two  expressions  for  work  there  results 
-ird2          ird 


or 


Now  the  volume  of  compressed  air,  v,  passing  through  the  pipe 
is,  in  cubic  feet, 


4  X  144 

and  the  volume  of  free  air  va  is  rv. 
Therefore 

W.2 

X  rs 


4  X  144 
and 

2  _  (4  X  144)  2t 


Insert  this  value  of  s2  in  Eq.  (25)  and  reduce  and  the  results 
4 


or 

/  =       ?  (26) 


FRICTION  IN  AIR  PIPES  51 

where  c  is  the  experimental  coefficient  and  includes  all  constants. 
From  Eq.  (26), 


**•(=£-)  (27) 

From  the  data^  recorded  in  the  appendix  the  coefficients  for 
formula  (26)  were  worked  out,  first  using  the  actual  measured 
diameters,  second  using  the  nominal  diameters.  The  average 
of  the  coefficients  for  each  size  pipe  were  then  platted  and  the 
results  tabulated  as  shown  on  Plate  II.  In  studying  this  plate 
it  should  be  borne  in  mind  that  the  vertical  scale  is  ten  times  that 
of  the  horizontal  which  exaggerates  the  irregularities  of  the 
coefficient. 

These  studies  reveal  conclusively  that  c  is  practically  independ- 
ent of  r  and  of  5  (the  velocity  in  pipes),  and  that  it  increases  as 
the  diameter  decreases.  If  temperature  has  any  effect,  it  could 
not  be  detected.  Since  the  friction  varies  inversely  as  the  fifth 
power  of  the  diameter,  it  is  very  sensitive  to  any  variation  in 
the  diameter.  Hence,  if  the  greatest  possible  accuracy  is  de- 
sired, the  computations  should  be  based  on  the  measured  diame- 
ter and  the  coefficient  taken  from  the  curve  AB,  Plate  II. 
If  the  actual  diameter  is  unknown  and  the  computer  must  use 
nominal  diameters,  the  coefficient  should  be  taken  from  the  line 
CD.  In  any  case  computations  of  friction  loss  in  commercial 
pipes  of  less  than  1  in.  in  diameter  will  be  unreliable  on  account 
of  the  relative  great  effect  caused  by  small  obstructions  and 
irregular  diameters. 

Table  IX  is  computed  from  Eq.  (26)  and  is  self-explanatory. 
It  affords  a  direct  and  easy  determination  of  friction  losses  in 
air  pipes. 

A  further  study  of  the  coefficients  found  by  the  curve  AB, 
Plate  II,  shows  that  the  logarithms  of  c  and  d  plat  to  a  straight 
line  from  which  is  obtained  the  relation 

_  0.1025 

C        ™~  i«       o  , 


This  inserted  in  Eq.  (26)  gives 

_  OL1026W 
J  -         rd^-zT~ 
or 

_  01025  Jvf 
'  3,600  rd5-31 


52 


COMPRESSED  AIR 


FRICTION  IN  AIR  PIPES 


Chart  for  Solving  Formula     /= 


rd6-31x3GOO 

/=  Friction  Loss  in  Pounds  per  Square  Inch. 
I  =  Length  of  Pipe  in  Feet. 
v=  Cubic  Feet  of  Free  Air  per  Minute. 
r  =  Ratio  of  Compression 
d=  Diameter  of  Pipe  in  Inches. 

The  Dependent  Factors   (fr)  ,   V   and  d    Lie  in 
a  Straight  Line.      To  get  the  Friction  Loss  in 
1000  Feet;   Divide  the  (fr)   by  r. 

Friction  of  Gasses   will  be  Proportional 
the  their  Specific  Gravities. 


-190 
-ISO 
-170 
-160 
-150 
-140 


-120 

-110 

100 

-  90 


_,  or  V2-  =  35.13  (fr}  d  6'81 


I 

I 


80,000 

ra 

65,000 
60,000 
55,000 
50,000 
45,000 
40,000 
35,000 
30.000 
28,000 
20.000 
24,000 
22  000 
20',  000 
18,000 
10.000 
14,000 
12.000 
10,000 
9000 
8000 
7000 
6000 


4000 
3500 
3000 


2000. 
1800 
1000 
1400 
1200 

1000 
900 
800 
700- 
600 

500- 
450- 
400- 
350- 
300- 

250- 
200- 

150- 


100. 
90- 


50- 
40- 


PLATE  III. 


15- 


10- 


0-1 


53 

12-1 


10  - 


5 

4H- 
4  - 

84- 
8- 

2H- 
2- 

m- 


m- 


54  COMPRESSED  AIR 

where  va  is  in  cubic  feet  per  minute.     Log   ,        .-  =  5.4544. 

" 


This  equation  gives  results  practically  indentical  with  those 
from  Eq.  (26)  when  c  is  taken  from  the  curve  AB.  It  is  almost 
as  easy  of  solution  and  has  the  advantage  that  it  is  independent 
of  a  table  of  coefficients. 

Plate  III  is  a  logarithmic  chart  for  solving  Eq.  (28a). 

Since  such  a  chart  can  handle  only  three  variables,  the  product 
fr  is  taken  as  a  single  variable  and  I  as  1,000  ft. 

To  solve  the  equation  by  this  chart,  lay  a  straight  edge  (or 
stretch  a  thread)  over  the  chart.  The  three  numbers  under 
the  line  will  satisfy  Eq.  (28a). 

Example  28a.  —  What  pressure  will  be  lost  in  a  4-in.  pipe 
5,000  ft.  long  when  transmitting  1,200  cu.  ft.  of  free  air  per 
minute  compressed  to  7  atmospheres  (r  =  7). 

A  thread  stretched  over  4  in.  and  1,200  cu.  ft.  crosses  the  fr 
line  at  25,  then  25  -4-  7  =  3.6  and  3.6  X  5  =  18  Ib. 

Since  the  process  of  designing  such  charts  as  Plate  III  has  not 
appeared  in  any  of  the  well-known  text-books,  the  author  has 
made  it  available  in  Appendix  B. 

The  following  formula  is  that  derived  by  Church  for  loss  by 
friction  in  air  pipes: 

2          2 
~ 


In  this  pz  and  pi  are  pressures  at  points  on  the  pipe  distance  Z 
apart,  pi  being  the  less  pressure,  A  is  the  area  of  the  pipe  and  c 
some  experimental  coefficient.  The  other  symbols  are  as  used 
elsewhere  in  this  article. 

Frank  Richards  recommends  a  simplification  of  Church's 
formula  by  assuming  c  constant  and  a  temperature  about  60°F. 
His  formula  is 

p*2  -  ^  =  *&w 

In  the  experiments  at  the  Missouri  School  of  Mines  in  1911 
(described  in  Appendix  C)  effort  was  made  to  find  the  laws  of 
resistance  to  flow  of  air  through  various  pipe  fittings.  Facili- 
ties were  not  available  for  sizes  above  2  in.  in  diameter  and  for  the 
smaller  sizes  the  results  were  erratic,  doubtless  due  to  the  rela- 
tively greater  effect  of  obstructions  and  variations  in  diameter 


FRICTION  IN  AIR  PIPES 


55 


in  the  small  pipes.     The  results  are  given  below.     Further  re- 
search is  needed  along  this  line. 

LENGTHS  OF  PIPE  IN  FEET  THAT  GIVE  RESISTANCE  EQUAL  THAT 
OP  A  SINGLE  FITTING 


Diameter 
of  pipe, 
inches 

Elbows  90° 

Unreamed 
joints,  2  ends 

Reamed  joints 

Return  binds 

Globe  valves 

H 

10.0 

2-4 

7 

10.0 

20 

H 

7.0 

2-4 

7 

7.0 

25 

i 

5.0 

2-4 

7 

5.0 

40 

ll/2 

4.0 

2-4 

7 

4.0 

45 

2 

3.5 

2-4 

7 

3.5 

47 

Tests  on  resistance  in  50-ft.  lengths  of  rubber-lined  armored 
hose,  with  their  end  fittings  such  as  is  used  to  connect  with  com- 
pressed-air tools,  were  made  with  average  result  as  follows : 


Diameter  of  hose,  inches  

H    * 

1 

l}4 

Resistance  in  50-ft.  length  

ao£ 

-T° 

,e£ 

Finally  it  is  important  to  note  that  in  cases  where  gases  other 
than  air  are  under  consideration  the  friction  losses  will  be  di- 
rectly proportional  to  the  specific  gravity  of  the  gas,  for  instance 
if  the  gas  has  a  specific  gravity  of  0.8  the  friction  will  be  0.8  of 
that  for  air  under  the  same  conditions. 

The  rate  of  flow  of  air  or  gas  through  a  long  pipe  of  uniform 
diameter  can  be  computed  approximately  by  observing  /  for 
distance  Z;  then 


in  case  of  air,  or 


Va    = 


in  case  of  gas  of  0.8  specific  gravity. 

This  formula  may  be  of  value  in  determining  the  flow  of  natu- 
ral gas  through  long  pipes. 

It  may  be  well  to  note  here  that  the  deposit  of  solid  matter 


56  COMPRESSED  AIR 

(paraffines  and  asphalts)  out  of  natural  gas  may  seriously  obstruct 
the  pipes  and  render  such  computations  altogether  inaccurate. 

Example.  —  1,600  cu.  ft.  per  minute  of  free  air  is  supplied  to  a 
mine  at  a  pressure  of  7  atmospheres  (r  =  7)  through  a  4-in. 
main.  At  a  distance  of  2,840  ft.  from  the  compressor  is  a  2-in. 
branch  placed  to  take  air  to  two  2J^  drills,  requiring  100  cu.  ft. 
each  of  free  air  per  minute.  The  2-in.  pipe  is  1,260  ft.  long  and 
has  in  that  length  two  globe  valves,  four  elbows,  and  eighteen 
unreamed  (extra)  joints. 

Each  drill  takes  its  air  through  50  ft.  of  1-in.  hose. 

What  will  be  the  loss  of  pressure  at  the  drills? 

Solution.  —  By  formula  (28a)  : 

Loss  in  the  4-in.  main: 


5.4544 
log  2,840  =  3.4533 
2  log  1,600  =  6.4082 

log  7  =  0.8457 
5.31  log  4  =  3.1972 

4.0423 

5.3159 
4.0423 

log/4  =  1.2736        .'.  /4  =  18.8  for  4-in.  pipe. 
Loss  in  2-in.  pipe.    Note  that  the  r  is  about  5.75  in  the  2-in  pipe  : 

Effective  length,  straight  1,260  ft. 

Effective  length,  2  glove  valves  @  47  94  ft. 

Effective  length,  4  elbows  @  3.5  14  ft. 

Effective  length,  18  unreamed  joints  3.0  54ft. 

Total  1,422  ft. 

5.4544  log  5.75  =  0.7597 

log  1,422  =  3.  1529  (5.31)  log  2  =  1.5983 

2  log     200  =  4.6020  2.3580 

3.2093 

2.3580 
log/2  =  0.8513         .'.  /2  =  7.10  for  2-in.  pipe. 

v  2 
Loss  in  50  ft.  of  1-in.  hose  delivering  100  cu.  ft.,  /  =  4.5  -£-  • 

Note  that  the  r  in  the  hose  is  (after  deducting  the  accumulated 
friction  in  the  4-in.  and  the  2-in.  pipes)  about  5.25. 


FRICTION  IN  AIR  PIPES  57 

log  4.5  =  0.  6532  log  5.25  =  0  .  7202 

2  log  100  =  4.0000          log  3,600  =  3.5563 
4.6530  4.2763 

4.2765 

log/  =  0.3765  .'.  2.4  for  the  50-ft.  hose. 

Total  loss  of  pressure  =  18.8  +  7.1  +  2.4  =  28.3. 

Evidently  such  computations  as  this  should  not  be  accepted  as 
giving  precise  results.  Such  matters  as  the  varying  r,  varying 
density  of  air  as  effected  by  temperature  and  free  air  pressure, 
irregular  qualities  and  changing  conditions  of  the  pipes,  leaks, 
and  irregular  demands  for  air  all  more  or  less  effect  the  resulting 
loss.  Nevertheless  such  computations  are  the  proper  guides  for 
the  designer. 

Art.  30.  Theoretically  Correct  Friction  Formula.  —  The  theo- 
retically correct  formula  for  friction  in  air  pipes  must  involve  the 
work  done  in  expansion  while  the  pressure  is  dropping. 

Let  pi  and  p2  be  the  absolute  pressures  at  entrance  and  dis- 
charge of  the  pipe  respectively  and  let  Q  be  the  total  weight  of 
air  passing  per  second. 

Then  the  total  energy  in  the  air  at  entrance  is 


and  at  discharge  the  energy  is 


The  difference  in  these  two  values  must  have  been  absorbed  in 
friction  in  the  pipe.  Hence,  using  the  expression  for  work  done 
in  friction  that  was  derived  in  Art.  29,  we  get 


£  -log*)  -       <*-., 

Numerical  computations  will  show  the  last  term,  viz., 


is  relatively  so  small  that  it  can  be  neglected  in  any  case  in 
practice  without  appreciable  error.  Hence,  by  a  simple  reduc- 
tion we  get 

i        Pi         irk      ,  dlrsz  .  ird2 

log<  £  =      «  x  IT  but  '•  =  4x15  rs' 


58  COMPRESSED  AIR 

which  when  substituted  gives 


2  I2pa  d 

or  considering  pa  as  constant, 


or 

logio  p2  =  logio  Pi  -  Ci  ^s2  (29) 

In  Eq.  (29)  c\  is  the  experimental  coefficient  and  includes  all 
constants,  s  is  the  velocity  in  the  air  pipe  and  varies  slightly 
increasing  as  the  pressure  drops.  All  efforts  so  far  have  failed 
to  get  a  formula  in  satisfactory  shape  that  makes  allowance  for 
the  variation  in  s. 

In  working  out  Ci  from  experimental  data  s  should  be  the  mean 
between  the  s\  and  s2,  and  when  using  the  formula  the  s  may  be 
taken  as  about  5  per  cent,  greater  than  si.  . 

Note  that  in  the  solution  of  Eq.  (29)  common  logarithms  should 
be  used  for  convenience,  allowing  the  modulus,  2.3  +  ,  to  go 
into  the  constant  c\. 

The  working  formula  may  be  put  in  a  different  and  possibly  a 
more  convenient  form,  thus.  In  the  expression 

i       irk  ^      dl       , 


substitute  for  s  its  value 

4  X  U4v 


and  reduce  and  we  get 

Iv  2 
log  p2  =  log  pi  -  c2  -—  (30) 


Still  another  form  is  gotten  thus.     The  whole  weight  of   air 
passing  is  va  X  wa=  Q,  and  by  Eq.  (13) 

Q  =  va  crfocj  and  therefore  va  =  - 


Oo.OOC  Pa 

Also 

Px  -,  Pa 

rx  =  —  and  wa  =    £      - 
pa  53.35* 

Substitute  these  in  (30)  and  it  reduces  to 


FRICTION  IN  AIR  PIPES  59 

(31) 


In  ordinary  parctice  —  may  be  taken  as  constant.     If  this  be 

wa 

done  Eq.  (31)  becomes 

logp2  =  logPl-c3^(^)2  (31a) 

a&  \px/ 

If  ta  =  525  and  wa  =  0.075,  then  c3  =  7,000  c2. 

In  (31)  and  (3 la)  px  varies  between  p}  and  p2.  Careful  com- 
putations by  sections  of  a  long  pipe  show  px  to  vary  as  ordinates 
to  a  straight  line.  Modifying  the  formulas  to  allow  for  this 
variation  renders  them  unmanageable.  In  working  out  the 
coefficient  px  may  be  taken  as  a  mean  between  p\  and  p2,  and  in 
using  the  formula  p  may  be  taken  as  p\  less  half  of  the  assumed 
loss  of  pressure. 

As  before  suggested,  common  logarithms  should  be  used  in 
all  the  equations  of  this  article. 

A  study  of  the  data  collected  in  Appendix  B  gives  values  for 
c2  Eq.  (31),  that,  for  pipes  3  to  12  in.  in  diameter,  conform  closely 
to  the  expression. 

c2  =  0.0124  -  0.0004d, 

which  gives  the  following: 

d"  =         3  4  5  6  8  10  12 

C2  =    0.0112  0.0108  0.0104  0.0100  0.0092  0.0084  0.0080 
C3  =      78.4       75.6       72.8       70.0       64.4       58.8       56.0 

With  these  coefficients  px  in  Eqs.  (31)  and  (3 la)  is  to  be  taken 
in  pounds  per  square  inch. 

Equations  (31)  and  (3 la)  are  theoretically  more  correct  than 
Eq.  (26)  and  the  coefficients  of  the  former  will  not  vary  so  much 
as  those  for  the  latter,  but  when  the  coefficients  are  correctly 
determined  for  Eq.  (26)  it  is  much  easier  to  compute  and  can  be 
adapted  to  tabulation,  while  Eq.  (31)  cannot  be  tabulated  in  any 
simple  way. 

Finally  it  should  be  said  that  extreme  refinement  in  computing 
friction  in  air  pipes  is  a  waste  of  labor,  for  there  are  too  many 
variables  in  practical  conditions  to  warrant  much  effort  at 
precision. 

Example  24a. — Apply  formulas  (26)  and  (31)  to  find  the  pres- 
sure lost  in  1,000  ft.  of  4-in.  pipe  when  transmitting  1,200  cu.  ft. 


60  COMPRESSED  AIR 

free  air  per  minute  compressed  to  150  gage  when  atmospheric 
conditions  are  pa  =  14.0,  wa  =  0.073  and  ta  =  540. 

Solution  by  Eq.  (20).—  r  =  15Q^  *  *  =  11.71.    By  Table  IX 

divide  23.44  by  11.71  and  the  result,  2  lb.,  is  the  pressure  lost 
per  1,000  ft. 

Solution  ofEq.  (31).  —  The  coefficient  for  a  4-in.  pipe  is  0.0108, 
and  log  pi  =  log  (150  +  14)  =  2.214844. 
Then 


l  nmno    54°     ^  1,000/1,200  w  0.073\  2 

log  p*  =  2.214844  -  0.0108  >   -  - 


The  log  of  the  last  term  is  3.791193  and  its  corresponding  number 
is  0.006183. 

2.214844  -  0.006183  =  2.208661  =  log  pz. 
Whence 

p2  =  161.7  +     and    pi  —  p2  =  2.3. 

Art.  31.  Efficiency  of  Power  Transmission  by  Compressed 
Air.  —  In  the  study  of  propositions  to  transmit  power  by  piping 
compressed  air,  persons  unfamiliar  with  the  technicalities  of 
compressed  air  are  apt  to  make  the  error  of  assuming  that  the 
loss  of  power  is  proportional  to  the  loss  of  pressure,  as  is  the  case 
in  transmitting  power  by  piping  water.  Following  is  the 
mathematical  analysis  of  the  problem: 

Pi  =  absolute  air  pressure  at  entrance  to  transmission  pipe, 
Pz  =  absolute  air  pressure  at  end  of  transmission  pipe, 
vi  =  volume  of  compressed  air  entering  pipe  at  pressure  p\, 
Vz  =  volume    of    compressed    air    discharged    from    pipe    at 
pressure  p*. 

Then  crediting  the  air  with  all  the  energy  it  can  develop  in 
isothermal  expansion,  the  energy  at  entrance  is  piVi  log  —  = 

Pa 

piUi  log  ri,  and  at  discharge  the  energy  is  p2vz  log  —  =  pzvz  log  r2. 

Pa 

Hence 


efficiency  E  =  =  (32) 


Common  logs  may  be  used  since  the  modulus  cancels.     The 
varying  efficiencies  are  illustrated  by  the  following  tables: 


FRICTION  IN  AIR  PIPES  61 

pa  =  14.5.     pi  =  87.     ri  =  6.     log  n  =  0.7781. 


7)0 

85 

80 

75 

70 

65 

60 

r2  

5.86 

5.52 

5.17 

4.83 

4.48 

4.14 

log  TZ 

0  7679 

0.7419 

0.7135 

0  .  6839 

0  6513 

0  6170 

E 

0  987 

0  953 

0  917 

0  879 

0  837 

0  793 

pa  =  14.5.     pi  =  145.     rj  =  10.     log 


1.000. 


•p2  . 

140 

135 

130 

125 

120 

r2         

9.66 

9.31 

8.97 

8.62 

8  28 

log  ?"  2         .          

0  .  9850 

0.9689 

0.9528 

0  9355 

0  9185 

E 

0  98 

0  97 

0  95 

0  93 

0  92 

The  above  examples  illustrate  the  advantage  in  transmitting 
at  high  pressure.  Of  course  the  work  cannot  be  fully  recovered 
in  either  case  without  expanding  down  to  atmospheric  pressure, 
and  to  do  this  in  practice  heating  would  be  necessary.  It  should 
be  understood  also  that  by  reheating  this  efficiency  can  be  exceeded. 

It  should  be  noted  also  that  the  above  does  not  apply  in  cases 
where  the  air  is  applied  without  expansion.  In  such  cases  the 
efficiency  of  transmission  alone  would  be 

(?2  -  Pa)  v*      ri(rz  -  1) 


(Pi  -  Pa)  vi       r2  (ri  -  1) 

Example  31a.  —  What  diameter  of  pipe  will  transmit  5,000 
cu.  ft.  of  free  air  per  minute  compressed  to  100  Ib.  gage,  with  a 
loss  of  10  per  cent,  of  its  energy,  in  2,500  ft.  of  pipe,  assuming 
pa  =  14.0? 

=          =  8.15;  then  by  Eq.  (30) 


Whence  log  r2  =  0.8200;  r2  =  6.6,  and  6.6  X  14  =  92.4. 

/  =  114  -  92.4  =  21.6  =  loss  of  pressure. 
By  Eq.  (27), 

log  d=\  [log  (0.06  X  2,500)  X  (^3^)  *-  log 

O  L  \    nu    / 


=  0.7602,  whence  d  =  5.75  in. 

Otherwise  go  into  Table  IX  with  loss  for  1,000  ft.  =  ^  =  8.64, 
and   8.64  X  r  =  8.64  X  7.37  =  63    (7.37   being  the  mean   r). 


62  COMPRESSED  AIR 

\ 

Then  opposite  5,000  in  the  first  column  find  nearest  value  to  63, 
which  is  55  in  the  6-in.  column;  showing  the  required  pipe  to  be 
a  little  less  than  6  in. 

Otherwise  over  Plate  III  stretch  a  thread  passing  over  63  on 
the  fr  line  and  5,000  on  the  Va  line.     It  will  cut  the  d  line  at 


Chart  for  Sohring  Formula     /_   .1025 1    Vs  or  vs-«  35^3  ffr\  4 

rd^xajOO 

/=  Friction  Loss  in  Pounds  per  Square  Inch.  ^ 

I  =  Length  of  Pipe  in  Feet.  75 

V  =  Cubic  Feet  of  Free  Air  per  Minute.  65 
i"—  Ratio  of  Compression 

d— Diameter  of  Pipe  in  Inches.  50 

The  Dependent  Factors  ( fr) ,   v   and  d   Lie  in  40 

a  Straight  Line.     To  get  the  Friction  Loss  in  35 

1000  Feet;  Divide  the  (fr)   by  r.  30 

Friction  of  G asses   will  be  Proportional 
the  their  Specific  Gravities . 


-160 
-140 


-110 
-100 


-70 

60 
55 


12 


200 


100 


16- 


PLATB  III. 


8H 


IK- 


i- 


r« 


(A) 


•*<r   10 

00ft  x 


000.31 

000.03s, 

Ooo.at  \              ... 

OVJ.Oi     \         >-'  i>:>* 

<XX>,dS                           G- 

ono.oe 
tii. 

M$ 

1 

ttuO.OC 

OOO.iiJI 

000.01 

000,  *I 

0»W.£I 

ooo  .oj 

.ooot 

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-000* 

. 

looei 

-0«M 

00*1 

•  owe; 

:% 

-OM 

. 

. 

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08- 

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-OC 


CHAPTER  IV 
OTHER  AIR  COMPRESSORS 

Art.  32.  Hydraulic  Air  Compressors.  —  Displacement   Type.  — 

Compressors  of  this  type  are  of  limited  capacity  and  low  effi- 
ciency, as  will  be  shown.  They  are  therefore  of  little  practical 
importance.  However,  since  they  are  frequently  the  subject 
of  patents  and  special  forms  are  on  the  market,  their  limitations 
are  here  shown  for  the  benefit  of  those  who  may  be  interested. 

Omitting  all  reference  to  the  special  mechanisms  by  which  the 
valves  are  operated,  the  scheme  for  such  compressors  is  to  admit 
water  under  pressure  into  a  tank  in  which  air  has  been  trapped 
by  the  valve  mechanisms.  The  entering  water  brings  the  air 
to  a  pressure  equal  to  that  of  the  water;  after  which  the  air  is 
discharged  to  the  receiver,  or  point  of  use.  When  the  air  is  all 
out  the  tank  is  full  of  water,  at  which  time  the  water  discharge 
valves  open,  allowing  the  water  to  escape  and  free  air  to  enter  the 
tank  again,  after  which  the  operation  is  repeated.  Usually  these 
operations  are  automatic.  The  efficiency  of  such  compression  is 
limited  by  the  following  conditions. 
Let  P  =  pressure  of  water  above  atmosphere,  or  ordinary  gage 

pressure, 
V  =  volume  of  the  tank. 

Then  P  +  Pa  =  absolute  pressure  of  air  when  compressed. 
The  energy  represented  by  one  tank  full  of  water  is  PV  and  by 
one  tank  full  of  free  air  when  compressed  to  P  +  pa  is 


Pa 

Therefore  the  limit  of  the  efficiency  is 

Pa  V  log,  r          pa  loge  T 

~PV~  ~P~ 

But  P  =  pi  —  pa)  where  pi  is  the  absolute  pressure  of  the  com- 
pressed air.  Inserting  this  and  dividing  by  pa  the  expression 
becomes 

E  =  loger  =  logic  rX  2.3 
r  -  1  r  -  1 

63 


64 


COMPRESSED  AIR 


Table  VII  is  made  up  from  formula  (33) . 
The  practical  necessity  of  low  velocities  for  the  water  entering 
and  leaving  the  tanks  renders  the  capacity  of  such  compressors 
low  in  addition  to  their  low  efficiency. 

Should  the  problem  arise  of  designing  a  large  compressor  of 
this  class  an  interesting  problem  would  involve  the  time  of  rilling 
and  emptying  the  tank  under  the  varying  pressure  and  head. 
Since  it  is  not  likely  to  arise  space  is  not  given  it. 

It  is  possible  to  increase  the  efficiency  of  this  style  of  com- 
pressor by  carrying  air  into  the  tank 
with  the  water  by  induced  current  or 
Sprengle  pump  action — a  well-known 
principle  in  hydraulics.  At  the  begin- 
ning of  the  action  water  is  entering  the 
tank  under  full  head  with  no  resistance, 
and  certainly  additional  air  could  be 
taken  in  with  the  water. 

Art.  33.  Hydraulic  Air  Compressors. — 
Entanglement  Type. — A  few  compressors 
of  this  type  have  been  built  compara- 
tively recently  and  have  proven  remark- 
ably successful  as  regards  efficiency  and 
economy  of  operation,  but  they  are 
limited  to  localities  where  a  waterfall  is 
available,  and  the  first  cost  of  installa- 
tion is  high. 

The  principle  involved  is  simply  the 
reverse  of  the  air-lift  pump,  and  what 
theory  can  be  applied  will  be  found  in 
Art.  39  on  air-lift  pumps. 

Figure  11  illustrates  the  elements  of 
a  hydraulic  air  compressor,  h  is  the 
effective  waterfall. 

H  is  the  water  head  producing  the  pressure  in  the  compressed  air. 
t  is  a  steel  tube  down  which  the  water  flows. 
S  is  a  shaft  in  the  rock  to  contain  the  tube  t  and  allow  the  water 

to  return. 
R  is  an  air-tight  hood  or  dome,  either  of  metal  or  of  natural  rock, 

in  which  the  air  has  time  to  separate  from  the  water. 
A  is  the  air  pipe  conducting  the  compressed  air  to  point  of  use. 


FIG. 


OTHER  AIR  COMPRESSORS  65 

b  is  a  number  of  small  tubes  open  at  top  and  terminating  in  a 
throat  or  contraction,  in  the  tube  t. 

By  a  well-known  hydraulic  principle,  when  water  flows  freely 
down  the  tube  t  there  will  occur  suction  in  the  contraction. 
This  draws  air  in  through  the  tubes  6,  which  air  becomes  en- 
tangled in  the  passing  water  in  a  myriad  of  small  bubbles;  these 
are  swept  down  with  the  current  and  finally  liberated  under  the 
dome  R,  whence  the  air  pipe  A  conducts  it  away  as  compressed 
air. 

The  variables  involved  practically  defy  algebraic  manipula- 
tion, so  that  clear  comprehension  of  the  principles  involved  must 
be  the  guide  to  the  proportions. 

Attention  to  the  following  facts  is  essential  to  an  intelligent 
design  of  such  a  compressor. 

1.  Air  must  be  admitted  freely — all  that  the  water  can  entangle. 

2.  The  bubbles  must  be  as  small  as  possible. 

3.  The  velocity  of  the  descending  water  in  the  tube  t  should 
be  eight  or  ten  times  as  great  as  the  relative  ascending  velocity 
of  the  bubble. 

The  ascending  velocity  of  the  bubble  relative  to  the  water 
increases  with  the  volume  of  the  bubble,  and  therefore  varies 
throughout  the  length  of  the  tube,  the  volume  of  any  one  bubble 
being  smaller  at  the  bottom  of  the  tube  than  at  the  top.  For 
this  reason  it  would  be  consistent  to  make  the  lower  end  of  the 
tube  t  smaller  than  the  top. 

Efficiencies  as  high  as  80  per  cent,  are  claimed  for  some  of 
these  compressors,  which  is  a  result  hardly  to  have  been  expected. 

The  great  advantage  of  this  method  of  air  compression  lies 
in  its  low  cost  of  operation  and  in  its  continuity.  Mechanical 
compressors  operated  by  the  water  power  could  be  built  for  less 
first  cost  and  probably  with  as  high  efficiency,  but  cost  of  opera- 
tion would  be  much  higher. 

Evidently  there  is  a  limit  to  the  amount  of  air  that  can  be 
taken  down  and  compressed  by  this  hydraulic  air  compressor. 
By  the  laws  of  conservation  of  energy  we  know  that  the  energy 
in  the  compressed  air  as  expressed  by  formula  pv  loge  r  cannot 
exceed  that  of  the  waterfall  which  is  Wh  where  W  is  the  weight 
of  water  passing,  or  in  general 

Wh 


The  limitation  can  also  be  seen  from  the  following  considerations: 

5 


66  COMPRESSED  AIR 

Let  V  represent  the  total  volume  of  air  in  the  whole  length  of 

the  downcast  pipe  t  and  let  A  represent  the  area  of  that  pipe. 

y 
Then  when  -r  =  h  the  downflow  of  water  will  cease,  for  the  static 

pressure  inside  and  outside  the  pipe  will  be  equal  —  in  this  state- 
ment friction  and  velocity  head  in  the  pipe  are  neglected.  A 
more  correct  statement  would  be  that  in  order  to  be  operative 


where  /  is  the  head  lost  in  friction  and  s  the  velocity  in  the 
downcast. 

Evidently  in  this,  V  is  the  dominant  number  and  it  can  be 
controlled  by  opening  or  closing  some  of  the  inlet  tubes  at  b. 
It  is  by  such  manipulation  that  the  most  efficient  working  can 
be  secured. 

Art.  34.  Centrifugal  and  Turbo  Air  Compressors.  —  With  the 
development  of  the  steam  turbine  it  has  become  practicable  to 
deliver  air  at  several  atmospheres  pressure  by  means  of  centrif- 
ugal machines. 

The  very  high  speed  at  which  such  machines  are  run  (up  to 
4,000  r.p.m.)  calls  for  the  most  perfect  possible  material  and 
workmanship.  Yet  they  are  relatively  simple,  occupy  small 
space,  are  of  low  first  cost  and  are  quite  efficient,  as  compared 
with  reciprocating  machines  to  do  equal  service.  These  quali- 
ties assure  this  class  of  machine  (which  includes  the  "  turbo  air 
compressors")  a  popularity  where  large  volumes  of  air  are  re- 
quired at  a  moderate  and  constant  pressure. 

One  very  effective  application  of  turbo  air  compressors  is  as 
a  "  booster"  to  large  reciprocating  machines,  the  scheme  being 
to  use  the  exhaust  steam  from  the  engines  to  run  the  steam  tur- 
bines that  actuate  the  turbo  compressors.  The  air  from  the 
turbo  compressors  is  delivered  into  the  intake  of  the  reciprocating 
machines.  A  relatively  small  increase  in  the  intake  pressure 
will  materially  influence  the  capacity  and  economy  of  opera- 
tion of  the  reciprocating  machines.  For  example:  Assume  that 
the  turbo  machines  deliver  air  at  ^2  atmosphere,  gage  pressure; 
that  is  r  =  \%.  Then  if  the  air  be  cooled  to  its  original  tem- 
perature before  entering  the  reciprocating  machine,  the  weight 
of  air  handled  will  be  increased  one-half.  Now  assume  the  re- 
ciprocating machine  to  have  been  designed  to  compress  free  air 


OTHER  AIR  COMPRESSORS  67 

to  a  ratio  r  =  6  or  about  75  Ib.  gage;  then  with  the  booster  at- 
tached, and  maintaining  the  same  ratio  (6)  of  compression 
within  the  compressor,  the  delivery  ratio  relative  to  atmosphere 
will  be  9  or  a  gage  pressure  about  120  Ib.  This  would  be  accom- 
plished without  compounding  and  without  development  of  any 
more  heat  than  without  the  booster.  However,  more  work 
would  be  required  of  the  reciprocating  engines.  Hence,  in 
studying  such  an  improvement  the  designer  should  determine 
whether  the  engines  can  meet  the  demand  for  increased  power. 
The  volume  of  air  delivered  by  and  the  efficiency  of  centrifugal 
and  turbo  compressors,  fans  and  blowers  are  matters  understood 
by  but  few,  seldom  known,  and  often  far  from  what  is  assumed  or 
claimed.  The  theory  underlying  these  subjects  is  somewhat 
difficult  and  is  deferred  to  Chapters  VIII  and  IX. 


CHAPTER  V 
SPECIAL  APPLICATIONS  OF  COMPRESSED  AIR 

IN  this  chapter  attention  is  given  only  to  those  applications  of 
compressed  air  that  involve  technicalities — with  which  the  de- 
signer or  user  may  not  be  familiar,  or  by  the  discussion  of  which 
misuse  of  compressed  air  may  be  discouraged  and  a  proper  use 
encouraged. 


Engine 


FIG.  12. 


Art.  35.  The  Return-air  System. — In  the  effort  to  economize  in 
the  use  of  compressed  air  by  working  it  expansively  in  a  cylinder 
the  designer  meets  two  difficulties:  first,  the  machine  is  much 
enlarged  when  proportioned  for  expansion;  second,  it  is  consider- 
ably more  complicated;  and  third,  unless  reheating  is  applied  the 
expansion  is  limited  by  danger  of  freezing. 

To  avoid  these  difficulties  it  has  been  proposed  to  use  the  air  at 

68 


SPECIAL  APPLICATIONS  OF  COMPRESSED  AIR  69 

a  high  initial  pressure,  apply  it  in  the  engine  without  expansion, 
and  exhaust  it  into  a  pipe,  returning  it  to  the  intake  of  the  com- 
pressor with  say  half  of  its  initial  pressure  remaining.  The 
diagram,  Fig.  12,  will  assist  in  comprehending  the  system. 

To  illustrate  the  operation  and  theoretic  advantages  of  the 
system  assume  the  compressor  to  discharge  air  at  200  Ib.  pressure 
and  receive  it  back  at  100  Ib.  Then  the  ratio  of  compression  is 
only  2  and  yet  the  effective  pressure  in  the  engine  is  100  Ib. 

Evidently  then  with  a  ratio  of  compression  and  expansion  of 
only  2  the  trouble  and  loss  due  to  heating  are  practically  removed ; 
and  the  efficiency  in  the  engine  even  without  a  cutoff  would  be, 
by  Eq.  (18)  72  per  cent.  By  the  above  discussion  the  advantages 
of  the  system  are  apparent,  and  where  a  compressor  is  to  run  a 
single  machine,  as  for  instance  a  pump,  the  advantage  of  this 
return-air  system  will  surely  outweigh  the  disadvantage  of  two 
pipes  and  the  high  pressure,  but  where  one  compressor  installa- 
tion is  to  serve  various  purposes  such  as  rock  drills,  pumps,  ma- 
chine shops,  etc.,  the  system  cannot  be  applied.  There  should 
be  a  receiver  on  each  air  pipe  near  the  compressor. 

Art.  36.  The  Return-air  Pumping  System. — Following  the 
preceding  article  it  is  appropriate  to  describe  the  return-air 
pumping  system.  The  economic  principle  involved  is  different 
from  that  of  the  return-air  system  just  explained. 

The  scheme  is  illustrated  in  Fig.  13.  It  consists  of  two  tanks 
near  the  source  of  water  supply.  Each  tank  is  connected  with  the 
compressor  by  a  single  air  pipe,  but  the  air  pipes  pass  through  a 
switch  whereby  the  connection  with  the  discharge  and  intake 
of  the  compressor  can  be  reversed,  as  is  apparent  on  the  diagram. 
In  operation,  the  compressor  discharges  air  into  one  tank,  thereby 
forcing  the  water  out  while  it  is  exhausting  the  air  from  the 
other  tanks,  thereby  drawing  the  water  in.  The  charge  of  air 
will  adjust  itself  so  that  when  one  tank  is  emptied  the  other  will 
be  filled,  at  which  time  the  switch  will  automatically  reverse  the 
operation. 

The  economic  advantage  of  the  system  lies  in  the  fact  that  the 
expansive  energy  in  the  air  is  not  lost  as  in  the  ordinary  dis- 
placement pump  (Art.  37).  The  compressor  takes  in  air  at  vary- 
ing degrees  of  compression  while  it  is  exhausting  the  tank. 

The  mathematical  theory,  and  derivation  of  formulas  for 
proportioning  this  style  of  pump  are  quite  complicated  but 
interesting. 


70 


COMPRESSED  AIR 


Preliminary  to  a  mathematical  study  for  proportioning  the 
installation  it  is  well  to  follow  a  cycle  in  its  operation :  Referring 
to  the  two  tanks,  Fig.  13,  as  A  and  B,  assume  tank  A  to  be  full 
of  air  at  a  pressure  sufficient  to  sustain  the  back  pressure  or  head 
of  the  discharge  water  column  and  tank  B  to  be  full  of  water. 
The  air  compressor  is  running  and  taking  the  compressed  air 
out  of  A  and  passing  it  over  into  B.  At  this  stage  (the  beginning 
of  a  cycle)  no  work  is  demanded  of  the  air  compressor  except 


=  Water  Supply 

FIG.  13. 


that  necessary  to  overcome  friction  in  the  air  and  water  pipes, 
but  as  the  air  is  exhausted  out  of  A  the  compressor  must  raise 
the  pressure  to  that  of  the  constant  water  head.  This  recom- 
pressed  air  goes  into  B  and  forces  the  water  out.  At  a  certain 
period  in  the  cycle  the  air  pressure  in  A  will  have  dropped  to  a 
point  when  water  will  begin  to  flow  in  through  the  intake  valves. 
After  this  point  in  the  cycle  we  may  assume  that  for  every  volume 
of  air  taken  out  of  tank  A  an  equal  volume  of  water  flows  in,  thus 
maintaining  a  constant  air  pressure  in  A  until  the  tank  is  filled 


SPECIAL  APPLICATIONS  OF  COMPRESSED  AIR  71 

with  water.  At  this  point  the  water  will  start  up  the  air  pipe  and 
a  sudden  drop  of  pressure  will  occur  in  the  intake  pipe  to  the  com- 
pressor. It  is  this  sudden  drop  that  is  utilized  to  operate  the 
reversing  switch,  which  completes  the  cycle. 

From  the  foregoing  it  becomes  evident  that  the  mathematical 
analysis  will  involve  the  matter  presented  in  Arts.  14  and  15, 
and  there  are  two  problems  to  solve  for  any  installation:  first, 
to  determine  the  piston  displacement  of  the  compressor  required 
to  deliver  a  specified  quantity  of  water  per  minute,  say,  second, 
to  design  the  steam  end  of  the  compressor  so  as  to  meet  the 
maximum  demand  for  power  which  occurs  once  in  each  cycle. 

The  first  problem  can  be  solved  by  Eq.  (17),  Art.  15,  which 
may  be  modified  thus: 

Let  va  =  the  actual  intake  capacity  of  the  compressor  (usually 
about  70  per  cent,  of  the  piston  displacement) ;  this 
may  be  taken  in  cubic  feet  per  minute. 

Let  m  =  number  of  minutes  required  to  bring  the  pressure 
down  from  po  to  pm. 

Then  by  Eq.  (17) : 

log  po  -  log  pm 

~   log  (V  +  Va)    ~   log  V' 

V  being  the  volume  of  one  tank  and  the  air  pipe  between  tank  and 
switch,  po  and  pm  being  the  highest  and  lowest  pressures  re- 
spectively occurring  in  a  tank  in  one  cycle.  If  a  tank  full  of  water 
(volume  V)  is  to  be  delivered  in  n  min.  the  time  n  measures  the 
length  of  a  cycle,  and  is  divided  into  two  parts:  first,  that  just 
noted  as  m;  and  second,  that  required  to  draw  the  tank  full  of 
water  after  water  begins  to  flow  in  under  pressure  p^-  This 

latter  is  —     Hence 

Va 

n  =  m  -\ 

Va 

The  solution  must  be  made  by  trial.  Thus  assume  va  and  find 
m,  then  n  by  the  equation  next  above.  Repeat  until  a  satis- 
factory n  has  been  found. 

The  second  problem  must  be  solved  by  the  matter  developed  in 
Art.  14.  There  it  is  shown  that,  with  sufficient  accuracy  for 
designing,  the  maximum  rate  of  work  occurs  when  r  =  3.  Hence, 
having  determined  va,  the  maximum  rate  of  work  demanded  of 


72  COMPRESSED  AIR 

the  steam  end  can  be  gotten  by  Eq.  (8)  or  Table  I  with  r  =  3, 
due  allowance  being  made  for  efficiency,  etc. 

Since  the  air  pipes  have  an  effect  analogous  to  the  clearance 
space  in  engines  they  should  be  made  small,  even  at  some  sacrifice 
in  friction.  A  velocity  of  40  to  50  per  second  may  be  allowed 
in  the  air  pipes. 

The  tanks  should  have  a  volume  not  less  than  ten  times  the 
volume  in  the  air  pipes. 

Theoretically,  the  pump  tanks  may  be  placed  above  the  water 
supply  as  shown  in  Fig.  13,  but  the  cycle  can  be  shortened  by 
submerging  the  tanks  and  thus  increasing  the  pressure  pm.  In 
any  case  where  the  tanks  are  to  be  submerged  the  valves  can 
still  be  placed  above  water  and  the  water  siphoned  over  into  the 
tanks  as  suggested  in  Fig.  13a. 

The  most  important  application  of  this  return-air  pumping 
system  has  been  in  pumping  slimes,  sand  and  acids — such  material 
as  would  soon  ruin  a  reciprocating  or  centrifugal  pump.  A 
number  of  such  pumps  are  in  use  pumping  cement  " slurry" 
which  is  a  fine-ground  alkaline  mud  or  slime  occurring  in  the 
process  of  manufacture  of  Portland  cement. 

The  agency  or  force  usually  utilized  for  automatically  operating 
the  switch  is  the  suction  or  partial  vacuum  in  the  intake  pipe 
which  (as  the  tanks  are  usually  installed)  will  suddenly  increase 
when  water  starts  up  the  air  pipe  as  described  above.  Other 
means  that  could  be  used  to  move  the  switch  are :  The  difference 
in  pressures  in  the  intake  and  discharge  pipes.  This  difference 
gradually  increases  until  the  switch  is  thrown  and  would  be  util- 
ized when  the  tanks  are  deeply  submerged.  Another  means 
would  be  to  switch  after  an  assigned  number  of  revolutions  of 
the  compressor — the  number  being  that  necessary  to  run  a  cycle 
as  described  above. 

The  switch  is  usually  made  in  the  form  of  a  piston  valve. 
Details  would  be  inappropriate  in  this  volume. 

Example  36. — Design  a  return-air  pump  to  deliver  200  cu.  ft. 
per  minute  under  a  head  of  100  ft.;  the  tanks  to  be  placed  at 
water  level  (as  in  Fig.  13a)  and  air  pipes  to  be  500  ft.  long. 

Solution. — The  ratio  po  -f-  pn  is  the  same  as  the  ratio  of  the 
water  heads  (taking  amtospheric  pressure  as  a  head  of  33.3  ft.). 
Then  this  ratio  is  133.3  -f-  33.5  =  4.  Assume  for  first  trial  that 
the  tanks  have  a  volume  of  400  cu.  ft.  each,  and  that  effective 


SPECIAL  APPLICATIONS  OF  COMPRESSED  AIR  73 


RETURN    AIR    PUMP 


FIG.  13a. 


74  COMPRESSED  AIR 

intake  capacity  of  the  compressor  is  1.5  cu.  ft.  per  stroke.  Then 
the  number  of  strokes  required  in  a  cycle  is 

1  I  '   n  =  I?  +  log  401.50-4  log  400  =  266  +  37°  -  636' 

636  strokes  =  318  revolutions  to  deliver  400  cu.  ft.,  or  159 
revolutions  to  deliver  200  cu.  ft.  per  minute. 

This  speed  is  excessive,  but  before  we  make  another  trial  we 
will  see  what  size  air  pipe  will  be  necessary  in  order  to  prescribe 
the  correct  size  of  tanks. 

159  X  2  X  1.5  =  477  cu.  ft.  per  minute  intake  to  the  com- 
pressor. This  is  the  maximum  rate  of  passage  of  air  through 
either  air  pipe  and  occurs  once  in  a  cycle,  and  that  just  at  the 
end  when  the  pressure  in  the  air  pipe  is  about  that  of  the  atmos- 
phere. It  is  at  this  time  that  friction  in  the  air  pipe  is  greatest 
and  we  may  allow  a  drop  (/)  of  5  Ib.  in  order  to  economize  in  size 
of  both  air  pipes  and  tanks.  Then  by  formula  (27  d) 


or  by  Plate  III  we  see  that  a  3K-in.  pipe  will  give  a  drop  of  4  Ib. 
in  500  ft.  The  volume  in  a  3K-in.  pipe  500  ft.  long  is  33.5  cu.  ft. 
Since  we  may  use  tanks  having  ten  times  the  volume  of  the  air 
pipe,  we  will  recalculate  for  tanks  335  cu.  ft.  and  a  compressor 
intake  capacity  of  2  cu.  ft.  per  stroke.  Then 


whence  402  strokes  or  200  revolutions  deliver  335  cu.  ft.  and 
120  would  deliver  200  —  which  is  about  the  right  speed  for  such 
a  compressor. 

It  remains  to  find  the  maximum  rate  of  work  required  of  the 
steam  end  of  the  compressor.  The  greatest  ratio  of  compression 
occurring  in  a  cycle  is  4,  but  by  Art.  14  we  know  that  the  great- 
est rate  of  work  occurs  when  the  ratio  is  about  3,  and  that  this 
max.  rate  is 


_  .  _  144  X         a  _ 

_L_-        (1.25)  » 


SPECIAL  APPLICATIONS  OF  COMPRESSED  AIR  75 

where  pQ  is  the  constant  delivery  pressure  in  pounds  per  square 
inch,  and  Va  is  the  effective  intake  capacity  of  the  compressor 
in  cubic  feet  per  minute  or  seconds  as  desired.  If  we  take  Va 
in  cubic  feet  per  second  and  divide  by  550  we  get  horsepower. 
Then  approximately  the  max.  horsepower  rate  is  one-tenth 
poVa.  This  is  a  general  rule  when  r  goes  up  to  3. 

Note  that  pQ  is  in  pounds  per  square  inch  and  that  n  is  not  the 
same  as  in  the  next  preceding  equation. 

Applying  this  to  the  numerals  above  we  get 

™  133.3  X  0.434  X  120  X  2  X  2 

Max.  horsepower  =  -  =  44. 

uu  X  IvJ 

A  description  of  the  installation  of  a  return-air  pump  and  a 
full  discussion  of  the  theoretic  design  can  be  found  in  Trans.  Am. 
Soc.  C.  E.,  vol.  54,  page  1,  Date,  1905. 

Art.  37.  Simple  Displacement  Pump. — First  Known  as  the 
Shone  Ejector  Pump. — In  this  style  of  pump  the  tank  is  sub- 
merged so  that  when  the  air  escapes  it  will  fill  by  gravity.  The 
operation  is  simply  to  force  in  air  and  drive  the  water  out.  When 
the  tank  is  emptied  of  water,  a  float  mechanism  closes  the  com- 
pressed-air inlet  and  opens  to  the  atmosphere  an  outlet  through 
which  the  air  escapes,  allowing  the  tank  to  refill.  Various 
mechanisms  are  in  use  to  control  the  air  valve  automatically. 
The  chief  troubles  are  the  unreliable  nature  of  float  mechanisms 
and  the  liability  to  freezing  caused  by  the  expansion  of  the  escap- 
ing air.  Some  of  the  late  designs  seem  reliable. 

The  limit  of  efficiency  of  this  pump  is  given  by  formula  (18) 
and  Table  VI.  The  pump  is  well  adapted  to  many  cases  where 
pumping  is  necessary  under  low  lifts.  In  case  of  drainage  of 
shallow  mines  and  quarries,  lifting  sewerage,  and  the  like,  one 
compressor  can  operate  a  number  of  pumps  placed  where  con- 
venient; and  each  pump  will  automatically  stop  when  the  tank 
is  uncovered  and  start  again  when  the  tank  is  again  submerged. 
See  page  120  for  design  of  a  system  of  such  pumps. 


CHAPTER  VI 
THE  AIR-LIFT  PUMP 

Art.  38. — The  air-lift  pump  was  introduced  in  a  practical  way 
about  1891,  though  it  had  been  known  previously,  as  revealed 
by  records  of  the  Patent  Office.  The  first  effort  at  mathematical 
analysis  appeared  in  the  Journal  of  the  Franklin  Institute  in 
July,  1895,  with  some  notes  on  patent  claims.  In  1891  the 
United  States  Patent  Office  twice  rejected  an  application  for  a 
patent  to  cover  the  pump  on  the  ground  that  it  was  contrary  to 
the  law  of  physics  and  therefore  would  not  work.  Altogether  the 
discovery  of  the  air-lift  pump  served  to  show  that  at  that  late 
date  all  the  tricks  of  air  and  water  had  not  been  found  out. 

The  air  lift  is  an  important  addition  to  the  resources  of  the 
hydraulic  engineer.  By  it  a  greater  quantity  of  water  can  be 
gotten  out  of  a  small  deep  well  than  by  any  other  known  means, 
and  it  is  free  from  the  vexatious  and  expensive  depreciation  and 
breaks  incident  to  other  deep  well  pumps.  While  the  efficiency 
of  the  air  lift  is  low  it  is,  when  properly  proportioned,  probably 
better  than  would  be  gotten  by  any  other  pump  doing  the  same 
service. 

The  industrial  importance  of  this  pump;  the  difficulty  sur- 
rounding its  theoretic  analysis;  the  diversity  in  practice  and 
results;  the  scarcity  of  literature  on  the  subject;  and  the  fact 
that  no  patent  covers  the  air  lift  in  its  best  form,  seem  to  justify 
the  author  in  giving  it  relatively  more  discussion  than  is  given 
on  some  better-understood  applications  of  compressed  air. 

Art.  39.  Theory  of  the  Air-lift  Pump. — An  attempt  at 
rational  analysis  of  this  pump  reveals  so  many  variables,  and  some 
of  them  uncontrollable,  that  there  seems  little  hope  that  a  satis- 
factory rational  formula  will  ever  be  worked  out.  However,  a 
study  of  the  theory  will  reveal  tendencies  and  better  enable  the 
experimenter  to  interpret  results. 

In  Fig.  14,  P  is  the  water  discharge  or  eduction  pipe  with 
area  a,  open  at  both  ends  and  dipped  into  the  water.  A  is  the 

76 


THE  AIR-LIFT  PUMP 


77 


air  pipe  through  which  air  is  forced  into  the  pipe,  P,  under  pres- 
sure necessary  to  overcome  the  head  D.  b  is  a  bubble  liberated 
in  the  water  and  having  a  volume  0  which  increases  as  the  bubble 
approaches  the  top  of  the  pipe. 

The  motive  force  operating  the  pump  is  the  buoyancy  of  the 
bubble  of  air,  but  its  buoyancy  causes  it  to  slip  through  the 
water  with  a  relative  velocity  u. 

In  one  second  of  time  a  volume  of  water  =  au  will  have  passed 
from  above  the  bubble  to  below  it  and  in  so  doing  must  have 
taken  some  absolute  velocity  s  in  passing  the  contracted  section 
around  the  bubble.  

Equating  the  work  done  by  the  buoyancy  of  the 
bubble  in  ascending,  to  the  kinetic  energy  given 
the  water  descending,  we  have 


wOu  =  wan  0     where  w 
*  9 


weight  of  water, 


or 


JL. 

2(7 


(a) 


2~  is  the  equivalent  of  the  head  h  at  top  of  the 
pipe  which  is  necessary  to   produce   s,   therefore 


is 


Suppose  the  volume  of  air,  0,  to  be  divided 
into  an  infinite  number  of  small  particles  of  air, 
then  the  volume  of  a  particle  divided  by  a  would  be  zero  and 
therefore  s  would  be  zero;  but  the  sum  of  the  volumes  (=  0) 
would  reduce  the  specific  gravity  of  the  water,  and  to  have  a 
balance  of  pressure  between  the  columns  inside  and  outside  the 
pipe  the  equation 

wO  =  wah 
must  hold. 

Hence  again  h  =  -,  showing  that  the  head  h    depends  only 

on  the  volume  of  air  in  the  pipe  and  not  on  the  manner  of  its 
subdivision. 

The  slip,  u,  of  the  air  relative  to  the  water  constitutes  the  chief 
loss  of  energy  in  the  air  lift.  To  find  this  apply  the  law  of  physics, 
that  forces  are  proportional  to  the  velocities  they  can  produce  in  a 
given  mass  in  a  given  time.  The  force  of  buoyancy  wO'  of  the 


78  COMPRESSED  AIR 

bubble  causes  in  1  sec.  a  downward  velocity  s  in  a  weight  of  water 
wau.     Therefore 

wO  _  s 

wau       g 
Whence 


Therefore 


Og      v  ,  0       s2 

u  —  _  _.     gut  _  _        ag  prcjved  above. 
as  a       20 


•£  =  •*/-£  (b) 


This  shows  that  the  slip  varies  with  the  square  root  of  the 
volume  of  the  bubble.  It  is  therefore  desirable  to  reduce  the 
size  of  the  bubbles  by  any  means  found  possible. 

o 

If  u  =  70  then  the  bubble  will  occupy  half  the  cross-section  of 

the  pipe.  This  conclusion  is  modified  by  the  effect  of  surface 
tension,  which  tends  to  contract  the  bubble  into  a  sphere. 
The  law  and  effect  of  this  surface  tension  cannot  be  formulated 
nor  can  the  volume  of  the  bubbles  be  entirely  controlled.  Unfor- 
tunately, since  the  larger  bubbles  slip  through  the  water  faster 
than  the  small  ones,  they  tend  to  coalesce  ;  and  while  the  conclu- 
sions reached  above  may  approximately  exist  about  the  lower  end 
of  an  air  lift,  in  the  upper  portion,  where  the  air  has  about  re- 
gained its  free  volume,  no  such  decorous  proceeding  exists,  but 
instead  there  is  a  succession  of  more  or  less  violent  rushes  of  air 
and  foamy  water. 

The  losses  of  energy  in  the  air  lift  are  due  chiefly  to  three 
causes:  first,  the  slip  of  the  bubbles,  through  the  water;  second, 
the  friction  of  the  water  on  the  sides  of  the  pipe;  and  third,  the 
churning  of  the  water  as  one  bubble  breaks  into  another.  As 
one  of  the  first  two  decreases  the  other  increases,  for  by  reducing 
the  velocity  of  the  water  the  bubble  remains  longer  in  the  pipe 
and  has  more  time  to  slip. 

The  best  proportion  for  an  air  lift  is  that  which  makes  the  sum 
of  these  two  losses  a  minimum.  Only  experiment  can  deter- 
mine what  this  best  proportion  is.  It  will  be  affected  somewhat 
by  the  average  volume  of  the  bubbles.  As  before  said,  any 
means  of  reducing  this  volume  will  improve  the  results. 

Art.  40.  Design  of  Air-lift  Pumps.  —  The  variables  involved 
in  proportioning  an  air-lift  pump  are  : 


THE  AIR-LIFT  PUMP 


79 


Q  =  volume  of  water  to  be  lifted,  per  second, 

h  =  effective  lift  from  free  water  surface  outside  the  discharge 

pipe, 
I  =  D  -\-  h  =  total  length  of  water  pipe  above  air  inlet, 

D  =  depth  of  submergence  =  depth  at  which  air  is  liberated  in 
water  pipe  measured  from  free  water  surface  outside  the 
discharge  pipe, 

va  =  volume  per  second  of  free  air  forced  into  well, 

a  =  area  of  water  pipe, 

A  =  area  of  air  pipe, 

0  =  volume  of  the  individual  bubbles. 

The  designer  can  control  A,  a,  D  +  h  and  va  but  he  has  little 
control  over  0,  and  cannot  foretell  what  D  and  Q 
will  be  until  the  pump  is  in  and  tested. 

When  the  pump  is  put  into  operation  the  free 
water  surface  in  the  well  will  always  drop.  What 
this  drop  will  be  depends  first  on  the  geology  and 
second  on  the  amount,  Q,  of  water  taken  out.  In 
very  favorable  conditions,  as  in  cavernous  lime 
stone,  very  porous  sandstone  or  gravel,  the  drop 
may  be  only  a  few  feet,  but  in  other  cases  it  may 
be  so  much  as  to  prove  the  well  worthless.  In  any 
case  it  can  be  determined  by  noting  the  drop  in 
the  air  pressure  when  the  water  begins  flowing. 
If  this  drop  is  p  lb.,  the  drop  of  water  surface  in 
the  well  is  2.3  X  p  ft. 

Unless  other  and  similar   wells  in  the  locality   A 
have  been  tested,  the  designer  should  not  expect 
to  get  the  best  proportion  with  the  first  set  of  pip-  p 
ing,  and  an  inefficient  set  of  piping  should  not  be 
left  in  the  well. 

The  following  suggestions  for  proportioning  air 
lifts  have  proved  safe  in  practice,  but,  of  course, 
are  subject  to  revision  as  further  experimental 
data  are  obtained  (see  Figs.  16  and  17). 

Air  Pipe.  —  Since  in  the  usually  very  limited  space  high  veloci- 
ties must  be  permitted  we  may  allow  a  velocity  of  about  30  ft.  per 
second  or  more  in  the  air  pipe. 

Submerge,  ice.  —  The  ratio  ^  ,  ,  is  defined  as  the  submergence 
ratio.  Experience  seems  to  indicate  that  this  should  be  not  less 


FIG.  15. 


80  COMPRESSED  AIR 

than  one-half;  and  about  60  per  cent,  is  a  common  rule  of 
practice.  Probably  the  efficiency  will  increase  with  the  ratio  of 
submergence,  especially  for  shallow  wells.  The  cost  of  the  extra 
depth  of  well  necessary  to  get  this  submergence  is  the  most 
serious  handicap  to  the  air-lift  pump. 

Ratio  -£-. — When  air  is  delivered  into  water  under  submergence 
D  its  ratio  of  compression  will  be 

_  D  +  33.3 
33T~~ 

33.3  ft.  being  a  fair  average  for  water  head  equivalent  to  one 
atmosphere. 

When  air  is  mixed  with  water  as  in  these  pumps,  it  may  be 
assumed  to  act  under  isothermal  conditions.  Then  the  energy 
in  the  air  is  pava  loge  r  while  the  effective  work  realized  by  water 
delivered  at  top  of  discharge  pipe  is 


s  being  the  velocity  of  discharge.     Write  ~  -  =  hi.     Then  if  E 

AQ 
be  the  efficiency  of  the  pump  (reckoned  from  energy  in  air  deliv- 

ered) we  have  the  equation 

wQ  (h+hi)  =  E  pa  valoger. 
Whence 

Va  _  w(h  +  hi)  ,    . 

Q   "  Epalo%e  r 

In  case  of  a  pure-water  air-lift  pump  w  =  62.5,  and  we  may 
take  for  average  atmospheric  conditions  pa  =  2,100.  Then 
multiplying  by  2.3  and  using  common  logarithms  the  formula 
becomes 

•n  Va  h    -f-    hi  /or\ 

For  pure  water  -  =  -—-— 


Complete  data  on  several  apparently  well-designed  air-lift 
pumps  with  ratio  of  submergence  between  50  and  65  per  cent. 
and  total  submergence  between  350  and  500  ft.  show  E  to  have  a 
value  between  45  and  50  per  cent,  (see  Art.  43). 

If  we  take  E  =  45  per  cent.,  Eq.  (35)  becomes 

va        h  +  hi 


Q       35  logib  r 


(36) 


THE  AIR-LIFT  PUMP 


81 


PLATE  IV. 


82  COMPRESSED  AIR 

Formula  (36)  is  recommended  for  the  design  of  deep-well 
pumps.  In  this  hi  may  be  taken  as  about  6  ft.  which  is  assum- 
ing a  discharge  velocity  between  20  and  25  ft.  For  shallow  wells 
hi  may  be  taken  as  1,  which  would  correspond  to  a  velocity  of 
8ft. 

The  curves  on  Plate  IV  represent  Eq.  (36)  for  ratios  of  sub- 
mergence as  shown  thereon.  Note  that  in  Plate  IV  an  efficiency 
of  45  per  cent,  is  assumed.  When  more  data  have  been  collected, 
some  modification  may  be  found  desirable. 

In  any  case  some  excess  air  capacity  should  be  provided;  for 
should  the  free  water  surface  in  the  well  drop  more  than  antici- 
pated, after  prolonged  pumping,  more  air  will  be  needed  to 
maintain  the  discharge.  This  is  apparent  on  Plate  IV.  Note 

that  as  submergence  ratio  S  decreases  -£  increases. 

v 

Velocity  in  the  Water  Pipe. — This  is  the  factor  that  most  affects 
the  efficiency,  but  unfortunately,  owing  to  the  usual  small  area  in 
the  well,  the  velocity  cannot  always  be  kept  within  the  limits 
desired.  The  complicated  action  and  varying  conditions  in  a 
well  make  the  designer  entirely  dependent  on  the  results  of  ex- 
perience in  fixing  the  allowable  velocities  in  the  discharge  pipes. 

The  velocity  of  the  ascending  column  of  mixed  air  and  water 
should  certainly  be  not  less  than  twice  the  velocity  at  which  the 
bubble  would  ascend  in  still  water.  This  would  probably  put  the 
most  advantageous  least  velocity  in  any  air  lift  at  between  5  and 
10  ft.,  and  this  would  occur  where  the  air  enters  the  discharge 
pipe. 

The  velocity  at  any  section  of  the  pipe  will  be 


a 

where  Q  and  v  are  the  volumes  of  water  and  air  respectively  and 
a  the  effective  area  of  the  water  pipe,  s  increases  from  bottom  to 
top  probably  very  nearly  according  to  the  formula 


a  \        r. 
where 

K  =  increment  of  velocity, 

r  =  ratio  of  compression  under  running  conditions, 
I  =  total  length  of  discharge  pipe  above  air  inlet, 
x  =  distance  up  from  bottom  end  of  air  pipe  to  section  where 
velocity  is  wanted. 


THE  AIR-LIFT  PUMP  83 

The  formula  is  based  on  the  assumption  that  the  volume  of 
air  varies  as  the  ordinate  to  a  straight  line  while  ascending  the 
pipe  through  length  I.  As  the  volume  of  each  bubble  increases 
in  ascending  the  pipe,  the  velocity  of  the  mixture  of  water  and  air 
should  also  increase  in  order  to  keep  the  sum  of  losses  due  to  slip 
of  bubble  and  friction  of  water  a  minimum;  but  for  deep  wells 
with  the  resultant  great  expansion  of  air  the  velocity  in  the  upper 
part  of  the  pipe  will  be  greater  than  desired,  especially  if  the  dis- 
charge pipe  be  of  uniform  diameter.  Hence,  it  will  be  advanta- 
geous to  increase  the  diameter  of  the  discharge  pipe  as  it  ascends. 
The  highest  velocity  (at  top)  probably  should  never  exceed  20 
ft.  per  second  if  good  efficiency  is  the  controlling  object. 

Good  results  have  been  gotten  in  deep  wells  with  velocities 
about  6  ft.  at  air  inlet  and  about  20  ft.  at  top  (see  Art.  43). 

Figure  16  shows  the  proportions  and  conditions  in  an  air  lift 
at  Missouri  School  of  Mines. 

The  flaring  or  slotted  inlet  on  the  bottom  should  never  be 
omitted.  Well-informed  students  of  hydraulics  will  see  the 
reason  for  this,  and  the  arguments  will  not  be  given  here. 

The  numerous  small  perforations  in  the  lower  joint  of  the  air 
pipe  liberate  the  bubbles  in  small  subdivisions  and  some  advan- 
tage is  certainly  gotten  thereby. 

No  simpler  or  cheaper  layout  can  be  designed,  and  it  has 
proved  as  effective  as  any.  It  is  the  author's  opinion  that 
nothing  better  has  been  found  where  submergence  greater  than 
50  per  cent,  can  be  had. 

Art.  41.  The  Air  Lift  as  a  Dredge  Pump. — The  possibilities 
in  the  application  of  the  air  lift  as  a  dredge  pump  do  not  seem 
to  have  been  fully  appreciated.  This  may  be  due  to  its  being 
free  from  patents  and  therefore  no  one  being  financially  interested 
in  advocating  its  use. 

With  compressed  air  available  a  very  effective  dredge  can  be 
rigged  up  at  relatively  very  little  cost  and  one  that  can  be  adapted 
to  a  greater  variety  of  conditions  than  those  in  common  use, 
as  the  following  will  show. 

Suggestions. — Clamp  the  descending  air  pipe  to  the  outside 
of  the  discharge  pipe.  Suspend  the  discharge  pipe  from  a  derrick 
and  connect  to  the  air  supply  with  a  flexible  pipe  (or  hose). 
With  such  a  rigging  the  lower  end  of  the  discharge  pipe  can  be 
kept  in  contact  with  the  material  to  be  dredged  by  lowering  from 
the  derrick;  the  point  of  operation  can  be  quickly  changed  within 


84  COMPRESSED  AIR 

the  reach  of  the  derrick,  and  the  dredge  can  operate  in  very 
limited  space.  In  dredging  operations  the  lift  of  the  material 
above  the  water  surface  is  usually  small,  hence  a  good  submer- 
gence would  be  available.  The  depth  from  which  dredging  could 
be  done  is  limited  only  by  the  weight  of  pipe  that  can  be  handled. 

In  case  of  air-lift  dredge  pumps  the  ratio  of  submergence  may 
be  large  and  the  weight  per  cubic  foot  lifted  will  be  greater  than 
62.5  Ib.  In  case  heavy  coarse  material  (such  as  gravel)  is  being 
lifted,  the  velocity  should  be  high. 

Though  the  author  has  found  no  data  by  which  the  efficiency 
of  such  pumps  can  be  determined,  the  following  example  is 
taken  for  illustration. 

Example  35. — What  is  the  ratio  -^  for  a  dredge  pump  when 

submergence  =  30  ft.,  net  lift  =  6  ft.,  velocity  at  discharge 
=  12  ft.,  percentage  of  silt  =  33.3  per  cent.,  weight  of  silt  = 
100  Ib.  per  cubic  foot,  and  efficiency  =  0.333? 

Solution.— Referring  to  Eq.  (34),  w  becomes  75,  hi  =  2, 
r  =  1.9  (submergence  30  ft.  in  pure  water).  Whence  we  get 

v±   =  75  X  8  +  (100  -  62.4)^  X  30 
Q   '      0.333  X  2,100  X  2.3  X  0.279 

Note  that  the  second  term  in  the  numerator  is  the  work  done  in 
lifting  the  silt  through  the  30  ft.  of  water. 

Art.  42.  Testing  Wells  with  the  Air  Lift.— The  air  lift  affords 
the  most  satisfactory  means  yet  found  for  testing  wells,  even  if 
it  is  not  to  be  permanently  installed.  Such  a  test  will  reveal,  in 
addition  to  the  yield  of  water,  the  position  of  the  free  water 
surface  in  the  well  at  every  stage  of  the  pumping,  this  being  shown 
by  the  gage  pressures.  However,  some  precautions  are  neces- 
sary in  order  properly  to  correct  the  gage  readings  for  friction 
loss  in  the  air  pipe. 

The  length  of  air  pipe  in  the  well  and  any  necessary  correc- 
tions to  gage  readings  must  be  known. 

The  following  order  of  proceeding  is  recommended. 

At  the  start  run  the  compressor  very  slowly  and  note  the  pres- 
sure p\  at  which  the  gage  comes  to  a  stand.  This  will  indicate 
the  submergence  before  pumping  commences,  since  there  will 
be  practically  no  air  friction  and  no  water  flowing  at  the  point 
where  air  is  discharged.  Now  suddenly  speed  up  the  compressor 
to  its  prescribed  rate  and  again  note  the  gage  pressure  pz  before 


THE  AIR-LIFT  PUMP 


85 


any  discharge  of  water  occurs.  Then  p2  —  pi  =  p/  is  the  pres- 
sure lost  in  friction  in  the  air  pipe.  When  the  well  is  in  full 
flow  the  gage  pressure  pa  indicates  the  submergence  plus  friction, 
or  submergence  pressure  ps  =  p3  —  p/.  The  water  head  in  feet 
may  be  taken  as  2.3  X  ps.  Then,  knowing  the  length  of  air 
pipe,  the  distance  down  to  water  can  be  computed  for  conditions 
when  not  pumping  and  also  while  pumping. 


FIG.  16. 

Art.  43.  Data  on  Operating  Air  Lifts. — In  Figs.  16  and  17  are 
shown  the  controlling  numerical  data  of  two  air  lifts  at  Rolla, 
Mo.  These  pumps  are  perhaps  unusual  in  the  combination  of 
high  lift  and  good  efficiency.  The  data  may  assist  in  designing 
other  pumps  under  somewhat  similar  circumstances. 

The  figures  down  the  left  side  show  the  depth  from  surface. 


86  COMPRESSED  AIR 

The  lower  standing-water  surface  is  maintained  while  the  pump 
is  in  operation ;  the  upper  where  it  is  not  working. 

The  broken  line  on  the  right  shows,  by  its  ordinate,  the  vary- 
ing velocities  of  mixed  air  and  water  as  it  ascends  the  pipe. 

The  pump,  Fig.  16,  delivers  120  gal.  per  minute  with  a  ratio 

f  T»£}£*      Q  1  T1 

— -7 —  =  6.0.     The  submergence  is  58  per  cent,  and  efficiency 

net  energy  in  water  lift 
=  -  -^r~  -  =  50  per  cent. 

pv  loge  r 

The  pump,  Fig.  17,  delivers  290  gal.  per  minute  with  a  ratio 

f  ]"£*£*    o  IT* 

7 —  =  5.2.     Submergence  =  64   per  cent,  and   efficiency  = 

net  energv  in  water  lift 

-^ —  -  =  45  per  cent. 

pv  loge  r 

The  volumes  of  air  used  in  the  above  data  are  the  actual 
volumes  delivered  by  the  compressor.  The  volumetric  effi- 
ciencies of  the  compressors  by  careful  tests  proved  to  be  about  72 
per  cent. 


CHAPTER  VII 

RECEIVERS  AND  STORAGE  OF  ENERGY  BY 
COMPRESSED  AIR 

Art.  44.  Receivers  for  Suppressing  Pulsations  Only. — Every 
air  compressor  of  the  reciprocating  type  has,  or  should  have,  an 
air  receiver  on  the  discharge  pipe  as  close  as  possible  to  the  com- 
pressor outlet.  The  chief  duty  of  this  receiver  is  to  absorb  or 
take  out  the  pulsations  caused  by  the  intermittent  discharge 
from  the  compressor  in  order  that  the  flow  of  air  through  the 
discharge  pipe  (beyond  the  receiver)  may  be  uniform,  a  condition 
evidently  essential  to  efficient  transmission.  Incidently  the 
receiver  serves  as  a  separator  for  some  of  the  oil  and  water  in  the 
air  and  as  a  store  of  compressed  air  that  may  be  drawn  from  when 
the  demand  is  temporarily  in  excess  of  the  compressor  delivery. 

There  is  no  standard  rule,  nor  can  there  be  one,  for  proportion- 
ing these  receivers.  However,  the  service  demanded  of  the  air 
will  usually  indicate  whether  or  not  a  large  receiver  is  desirable. 
The  least  size  would  apply  in  cases  where  the  use  of  the  air  is 
continuous  and  the  needed  pressure  constant — as  in  air-lift 
pumps.  In  such  cases  the  requirement  of  the  receiver  is  solely 
to  suppress  the  pulsations.  For  such  cases  a  rational  formula 

for  the  volume  of  the  receiver  would  be  V  =  c  -  in  which  V  is 

r 

the  volume  of  the  receiver,  v  the  piston  displacement  of  one  stroke 
(of  low-pressure  cylinder),  r  the  total  ratio  of  compression,  and 
C  an  empyrical  coefficient  fixed  by  experiment  or  observation. 

Observe  that  v  -r-  r  is  the  volume  of  compressed  air  per  stroke, 
which  being  suddenly  discharged  causes  the  pulsations.  The 

question  remains,  what  ratio  of  V  to  -  will  be  necessary  to  suffi- 
ciently suppress  the  pulsations?  The  author  finds  no  light  on  the 
subject  in  compressed  air  literature.  Practice  does  not  seem  to 
distinguish  between  this  case  and  the  more  usual  one  where 
some  storage  capacity  is  needed.  The  author  is  of  the  opinion 
that  in  such  cases  (the  air  lift  for  instance)  a  much  smaller  re- 

87 


88  COMPRESSED  AIR 

ceiver  could  be  used  without  detrimental  effect,  and  thereby  the 
cost  reduced. 

Art.  45.  Receivers.  Some  Storage  Capacity  Necessary. — 
In  the  majority  of  compressed-air  installations  the  use  of,  or 
demand  for,  air  will  not  be  constant,  as  for  instance  in  machine 
shops,  quarries,  mines,  etc.  In  any  such  case  the  use  of  air  may 
exceed  the  compressor  capacity  for  a  short  time  and  then  for  a 
time  may  not  be  as  great  as  the  compressor  capacity.  In  these 
(the  more  common)  cases  the  receiver  is  intended  to  serve  as  a 
storage  reservoir  in  addition  to  its  several  other  duties. 

The  problem  of  determining  the  necessary  volume  for  the  stor- 
age is  simple,  provided  the  maximum  rate  of  use  and  its  duration 
in  time  can  be  gotten;  but  this  is  seldom  possible  as  will  be  readily 
conceded  when  the  complexity  and  irregularity  of  the  service 
is  considered. 

However,  the  problem  may  be  better  understood  if  expressed 
as  a  formula: 

Let  V  =  volume  of  receiver,  or  storage  reservoir,  in  cubic  feet, 
va  =  free-air  capacity  of  compressor  in  cubic  feet  per  min., 
Pi  =  highest  pressure  (absolute)  permitted  in  the  system, 
pz  =  lowest  satisfactory  working  pressure  permissible, 
R  =  maximum  rate  of  usage,  cubic  feet  free  air  per  min., 
T  =  duration  in  minutes  through  which  R  is  continued. 

To  get  a  simple  and  sufficiently  approximate  relation  assume 
isothermal  changes  and  equate  the  pressure  by  volume  products 
thus: 

Pl7  +   TpaVa   =   P*V  +   TRpa. 

Whence, 

V  =       Pa       T(R  -  va) 
Pi  -  Pz 

Example. — A  shop  has  a  compressor  with  va  =  300  and  normal 
pressure  pi  =  100  ft.  A  drop  of  10  Ib.  (p2  =  90)  is  permissible 
when  the  demand  is  double  the  average  for  2  minutes.  Then 

V  =  ^  X  2  X  (600  -  300)  =  900. 

A  calculation,  such  as  above,  applied  to  the  more  common  in- 
stallations, will  show  the  desirable  receiver  capacity  much  greater 
than  is  installed. 

The  common  practice   seems  to  be  to  install  a  compressor 


RECEIVERS  AND  STORAGE  OF  ENERGY  89 

capable  of  meeting  the  maximum  demand  without  storage,  and 
then  let  it  run  idle  much  of  the  time.  Going  to  this  extreme  is 
profitable  for  the  compressor  makers,  but  expensive  to  the  user 
in  first  cost  of  the  compressor  and  still  more  so  in  the  continual 
cost  of  extra  fuel  to  run  the  larger  compressor  even  though  idle. 

Where  a  compressor  has  been  installed  with  inconsiderable 
receiver  (or  storage)  capacity  and  the  business  outgrows  the 
capacity  of  the  compressor  as  thus  equipped,  the  addition  of  a 
considerable  storage  volume  may  defer  the  time  for  purchase  of 
a  larger  compressor  for  several  years,  and  at  the  same  time  get 
the  needed  additional  air  more  economically  than  if  a  larger 
compressor  were  installed.  This  claim  of  economy  is  based  on 
the  fact  that  a  small  machine  running  more  continually  and  with 
a  nearly  constant  load  is  more  economical  than  a  larger  machine 
running  constantly  but  with  intermittent  load.  The  case  is 
analogous  to  the  use  and  duty  of  a  distributing  reservoir  in  a 
water-supply  system. 

Art.  46.  Hydrostatic  Compressed-air  Reservoirs. — In  cases 
where  it  is  desired  to  store  large  units  of  energy  in  the  form  of 
compressed  air,  and  that  energy  is  to  be  drawn  out  with  but  little 
drop  in  the  air  pressure,  a  computation  of  the  volume  necessary 
for  tanks  under  conditions  heretofore  assumed  is  discouraging. 

Example. — Assume  that  storage  is  to  be  provided  for  air  at 
125  Ib.  (absolute)  such  that  100  hp.  can  be  drawn  from  the  storage 
for  1  hr.  with  a  drop  in  pressure  to  100  Ib.  What  volume  of 
storage  is  needed?  For  this  example  assume  all  changes  to  be 
isothermal.  * 

lOK  IClft 

Then  100  X  33,000  X  60  =  paVa  loge^  -  paVJoge^ 

=  2,116  Va  (2.140  -  1.917), 

whence  Va  =  4,200,000  and  V  =  50,000  approximately.  Sup- 
pose now  that  all  the  compressed  air  in  volume  V  at  pressure 
125  can  be  used  without  any  reduction  of  pressure.  What 
volume  would  give  100  hp.  for  1  hr.? 

Then  100  X  33,000  X  60  =  paVa  log  8.5, 

whence  Va  =  43,700  and  V  =  5,150  or  about  one-tenth  of  that 
required  under  the  first  assumption. 

This  latter  condition  (making  the  entire  volume  of  compressed 
air  available  without  reduction  of  pressure)  can  be  accomplished 


90 


COMPRESSED  AIR 


simply  and  economically  by  the  scheme  illustrated  by  Fig.  18 
which  needs  but  little  explanation. 

The  water  head  against  the  air  may  be  assumed  constant. 
The  dip  down  in  the  water  pipe  below  the  air  reservoir  is  to 
prevent  blowout  through  the  water  pipe  should  all  the  water  be 
forced  out  of  the  air  reservoir.  A  popoff,  or  an  automatic,  stop 
for  the  compressor,  would  be  adjusted  to  act  when  the  water  line 
dropped  down  below  the  air  tank  as  at  C.  Evidently  the  water 
pipe  ABC  need  not  be  vertical  nor  in  a  vertical  plane.  The  water 
reservoir  can  be  economically  placed  on  a  hilltop  in  the  neighbor- 


FIG.  18. 

hood,  or  the  air  reservoir  can  be  placed  in  underground  chambers 
(abandoned  chambers  in  mines,  for  instance)  and  the  water 
reservoir  at  the  surface. 

This  last  suggestion  naturally  leads  to  that  of  using  under- 
ground chambers  naked,  that  is,  without  the  steel  tanks.  This 
is  quite  feasible.  To  make  the  walls  of  such  a  chamber  tight 
against  escape  of  air  into  the  rock  the  "cement-gun"  is  ideal. 
Note  that  there  is  no  necessity  for  smooth  or  even  surfaces. 
The  cement-gun  may  be  found  more  efficient  if  used  while  the 
chamber  is  under  some  pressure.  The  cement  will  thus  be  driven 
into  every  crevice  and  pore  into  which  air  may  tend  to  escape. 


CHAPTER  VIII 
FANS 

Art.  47. — The  discussion  in  this  article  will  apply  to  any 
centrifugal  pump  and  to  fans  of  the  low-pressure  type  such  as 
are  applied  in  ventilation  of  mines  and  buildings,  when  change 
in  density  of  the  air  may  be  neglected. 

Though  the  discussion  is  brief,  the  student  entering  the  subject 
for  the  first  time  will  find  some  difficulty  in  keeping  in  mind  the 
several  qualifying  conditions  such  as  relative  velocities,  velocity 
heads,  pressures  within  and  without  the  wheel,  etc.  He  is  warned 
not  to  jump  at  conclusions  in  this  nor  any  other  branch  of  fluid 
mechanics,  but  is  urged  to  study  and  review  each  demonstration 
over  and  over  again  until  familiar  with  it.  We  hope  to  encourage 
an  interest  in  this  subject  by  saying  beforehand  that  there  are 
several  fallacious  opinions  that  can  be  successfully  contended 
with  only  by  those  who  are  well  grounded  in  the  following  under- 
lying theory. 

We  will  first  study  the  theory  as  revealed  by  the  laws  of  pure 
mechanics  and  the  conservation  of  energy,  without  considering 
the  effects  of  friction,  imperfection  of  design  or  improper  opera- 
tion. Formulas  thus  obtained  will  not  closely  check  with  re- 
sults from  a  pump  or  fan  in  operation,  but  they  tell  what  per- 
fection would  be,  and  so  show  how  far  short  we  fall  in  practice; 
and  they  point  to  the  best  lines  of  improvement  in  design  and  in 
operation. 

In  addition  to  the  symbols  shown  on  Fig.  19,  the  following  will 
be  used : 

w  =  weight  of  cubic  unit  of  fluid, 

b  =  width  of  discharge  at  outer  limit  of  vanes, 

bi=  width  of  inlet  at  inner  limit  of  vanes, 

Q  =  volume  of  fluid  passing,   cubic  feet  per  second  unless 

otherwise  stated, 

W  =  total  weight  of  fluid  passing  per  second  wQ, 
p  =  pressure  head  immediately  after  escape  from  revolving 

wlieel, 

H  =  total  head  given  to  the  fluid  by  the  wheel. 

91 


92 


COMPRESSED  AIR 


The  reason  for  using  heads  instead  of  pressures  is  that  the 
formulas  are  thereby  simplified.     Note  that  head  must  be  in 


J       V 


C'      <dx  B' 


-^f^- 

6 

(0 


(f) 


FIG.  19. 

feet  of  the  fluid  passing.     In  case  of  air  the  air  head  is  to  be  of 
constant  density. 


FANS  93 

Art.  48.  Purely  Centrifugal  Effects. — Consider  a  prism  of  fluid 
of  unit  area  and  extending  between  the  limits  r\  and  r,  in  a 
revolving  wheel  without  outlet,  as  CB,  Fig.  19.  The  centrifugal 
force  of  an  elementary  disk  across  this  prism,  of  thickness,  dx, 
and  distance  x}  from  the  axis  of  revolution,  is  by  well-known  laws 
of  mechanics 

,,.      wux2dx 
df  =  - 

gx 

where  ux  is  the  velocity  of  revolution  at  the  distance  x  from  the 
center.     Thus 


and  therefore 


x 
ux  =  -  u 


,.      wu2     , 
df  =  — r  xdx 

gr2 


and  the  total  centrifugal  force/,  which  is  effective  at  the  outer  end 
of  this  prism,  is  the  integral  between  the  limits  x  =  r  and  x  =  r\. 
Hence 

wu2    ,  0  w   ,  0 


since 

u\  =  —  u. 

This  is  the  pressure  on  a  unit  area  at  the  circumference  of  the 
wheel,  and,  evidently,  it  is  independent  of  the  form  or  cross- 
section  of  the  arm  CB.  Now,  pressure  divided  by  weight  gives 
head.  Hence,  the  pressure  head  against  the  walls  of  the  wheel 
at  the  circumference  is 

,.       u2  -  Ul2 

h1  =  — ^ 

u2 
Note  that  if  TI  =  0  then  h  =  ~ — 

Note  that  this  h  does  not  include  velocity  of  rotation. 

Now,  if  an  orifice  be  opened  at  the  circumference,  in  any  di- 
rection whatever,  and  the  pressure  outside  be  the  same  as  at  the 
entrance,  the  velocity  of  the  discharge,  relative  to  the  revolving 
walls  of  the  wheel,  will  be 

V  =  \/2  gh  or  V2  =  u2  —  Wi2. 

Note  that  if  r  =  0,  V  =  u. 

Note  also  that  the  absolute  velocity  v  of  discharge  is  made  up 


94  COMPRESSED  AIR 

of  the  two  components  V  and  u  and  in  amount  v2  =  u2  +  V 
+  2  uV  cos  0  =  2  u2  +  2  wF  cos  /3  when  n  =  0. 

v2 
Total  head.  #,  in  the  departing  fluid  =  ^—  or 


(37) 


When  there  is  a  discharge,  there  must  be  an  initial  velocity, 
Vi,  at  the  entrance,  and  this  must  be  considered  in  the  final 
head  within  the  wheel.  Thus,  the  total  relative  head  at  B  will 
now  be 

,        Fi2   .   u2  -  u,2 

=  20~  rt^ 

and  the  velocity  of  the  discharge,  relative  to  the  revolving  parts, 
will  have  the  relation 


Suppose,  now,  that  CB  is  a  radial  frictionless  tube,  open  at 
both  ends,  and  that  a  particle  of  matter  starts  from  a  state,  Vi, 
relative  to  the  tube,  and  moves  out,  without  change  of  pressure, 
from  radius  TI  to  r,  in  obedience  to  the  laws  of  centrifugal  force 
(or  acceleration).  Its  radial  acceleration,  when  distant  x  from 
the  center,  is,  by  well-known  laws  of  mechanics: 

Acceleration 

ux2  =  dVx 
x          dt  ' 
Also 

v     -  —    ' 
Vx  ~  dt' 

Therefore,  by  eliminating  dt,  we  get 
but,  as  before, 


x 
ux  =  —  u 


(sub  x  indicating  the  conditions  at  the  distance  x  from  the  cen- 
ter).    Therefore, 

V*dV*  =        xdx. 


FANS 


95 


Integrating  between  the  limits  V  and  Fi,  on  one  side,  and  r  and 
ri,  on  the  other,  we  get  as  before 


f.2    - 


(I) 


Art.  49.  Impulsive  or  Dynamic  Effects. — We  have  now  to 
study  the  effect  of  picking  up  the  fluid  at  entrance  to  the  moving 
parts  of  the  wheel.  This  will  be  studied  by  a  method  somewhat 
different  from  that  preceding: 

Assuming  the  fluid  to  be  at  rest  until  influenced  by  the  wheel, 
we  see  that  during  each  second  there  is  a  weight,  W,  brought  to  a 


FIG.  20. 

velocity,  v,  Fig.  20.     Now  the  reaction  against  the  wheel  due  to 

Wv 
the  creation  of  the  velocity,  v,  is  F  —  —  and  the  component  of 

y 

this  velocity  opposite  in  direction  to  the  rotation,  u,  is  v  cos  8 
and  this  equals  u  —  V  cos  (180°  —  0)  =  u  +  V  cos  0  and  since 
work  is  force  multiplied  by  distance,  the  work  done  in  overcoming 
the  reaction  is 

W  W 

u(u  +  V  cos  0)  —  =  (u2  +  uV  cos  |8)  —  • 
i  y  y 

If  H  be  the  total  head  given  the  fluid  up  to  the  point  considered, 


96  COMPRESSED  AIR 

then  work  done  =  WH,  since  all  the  head  has  been  imparted  by 
the  wheel. 
Hence, 

H  =  U*  +  UV-C^  (37) 

Note  that  it  is  preferable  to  use  the  angle  j8  rather  than  a  for  /3 
is  fixed  and  is  an  element  in  the  design  of  the  machine,  while  a 
varies  with  u  and  V. 

The  demonstration  above  evidently  applies,  however  short  the 
radial  depth  of  the  vanes  (r  —  n).  So  we  may  say  that  it 
applies  at  the  entrance  where  r  —  n  =  0.  Here,  then,  we  find  Eq. 
(37)  applies  in  case  of  purely  impulsive,  or  dynamic  action  with 
neither  centrifugal  force  nor  centrifugal  acceleration. 

It  is  now  apparent  that  no  matter  at  what  distance  from  the 
center  of  rotation  the  fluid  is  engaged  by  the  wheel,  it  will  have 
imparted  to  it  a  head  the  same  as  if  it  had  been  under  influence  of 
the  wheel  from  the  center  out. 

Art.  50.  Discharging  against  Back  Pressure.  —  Note  that  so 
far  we  have  assumed  that  the  pressure  against  the  discharge  is 
the  same  as  at  intake.  Under  this  condition  the  relative  velocity 
of  escape  will  be  V  =  u,  no  matter  in  what  direction  V  may  be, 
relative  to  the  wheel. 

We  have  now  to  establish  a  general  formula  for  H  when  pres- 
sure head  against  outlet  is  p,  and  at  inlet  p\.  Note  that  pi  may 
be  and  generally  is  negative  in  centrifugal  pumps,  but  in  fans 
it  is  usually  zero. 

We  have  established  the  fact  that  the  pressure  head  developed 

u2 
within  the  wheel,  when  no  discharge  is  allowed,  is  h  =  ~-  '     Now 

if  an  orifice  be  opened  through  the  periphery  of  the  wheel  into 
the  discharge  duct  where  pressure  head  =  p,  the  velocity  of 
escape  (relative)  will  be 


~  +  pi-p    =  u*  +  2g(Pl  -  p)  (II) 

Whatever  the  absolute  velocity  of  escape  v  may  be,  the  total 
absolute  head  added  to  the  fluid  by  the  machine  will  be 

a  =  f+p-pi  (in) 

«  Q 

Note  that  when  pi  is  negative  (or  suction),  it  becomes  a  plus 
addition  in  (III). 


FANS  97 

From  pure  trigonometry  we  have  in  any  case 

^2  =  ui  +  72  _j_  2  uV  cos  0. 

Now  in  (III)  replace  z;2  from  the  last  expression,  then  replace 
V2  with  its  value  from  (II)  and  we  get  as  before 

H  =  tt2  +  ^C°sg  (37) 

We  have  now  proven  Eq.  (37)  to  be  correct  for  both  purely 
centrifugal  and  impulsive  action,  and  to  be  independent  of 
entrance  and  discharge  pressures. 

Equations  (II),  (III)  and  (37)  are  the  theoretic  relations  when 
effects  of  friction  and  imperfections  of  design  are  neglected. 
Results  in  practice  may  be,  and  often  are,  quite  different  from 
what  this  theory  would  indicate,  due  to  imperfections  of  design, 
some  of  which  cannot  be  overcome  entirely. 

v2 

Note  that  if  p\  =  p,  then  H  =  ^-  and  V  =  u;  and  if  pi  =  p 

*9 

u2 
and  0  =  90°  then  v2  =  2u2.     When  0  =  90°,  #  =  —  irrespective  of 

u2 
pressures.     Also,  when  0  is  less  than  90°,  H  is  greater  than  — 

u2 
and  when  0  is  greater  than  90°,  H  is  less  than  —  * 

The  pressure  that  a  pump  or  fan  can  produce  depends  on  H. 
P  =  wH  when  the  whole  energy  is  transformed  into  pressure 
head.  Otherwise,  in  general 


at  or  near  outlet,  friction  being  neglected. 

The  work  required  of  the  machine  (neglecting  friction,  etc.) 
is  WH  =  wQH  where  W  and  Q  are  total  weight  and  total  volume 
passed,  respectively,  and  this  will  in  actual  performance  nearly 
equal  the  theoretic,  regardless  of  friction  and  other  losses. 

u2 
Note  that  V  may  be  zero  and  still  H  =  —  .     In  this  case  the 

u2 
fluid  revolving  in  the  wheel  has  a  pressure  head  =  s~,  and  a  ve- 

u2 
locity  head  =  ^~,  the  total  head  being  the  sum.     In  this  case 

the  work  is  zero  since  W  =  0.     If  the  pressure  head,  p,  in  the 

u2 
discharge  duct  be     -,  there  will  be  no  discharge,  the  pressure 


98  COMPRESSED  AIR 

inside  and  outside  the  wheel  balancing.  As  p  decreases  V  in- 
creases (see  Eq.  (II))  and  therefore  also  Q  increases. 

In  case  of  pumps  when  0  is  less  than  90°  the  pump  cannot  start 
a  discharge  under  full  back  pressure,  but  if  0  be  less  than  90° 
the  pump  may  start  under  its  full  head. 

Art.  51.  Designing. — The  dimensions  of  any  pump  or  fan  must 
conform  to  the  following  formulas,  which  hold  in  all  cases: 

Q  =  2  TrrbV  sin  0  =  2  nrb  sin  ft  \u2  -  2  g(pl  -  p)         (7«) 
Also  Q  =  2  TrribiVi  sin  <f>  (Vb) 

Note  that  V  sin  ft  and  V\  sin  6  are  the  radial  components  of 
velocity  of  discharge  at  outlet  and  inlet  respectively. 

In  designing  a  fan  or  pump,  the  chief  factors  are  H  and  Q. 
By  equation  (37)  these  factors  are  seen  to  be  interdependent 
(except  where  ft  =  90°),  since  for  any  completed  machine  Q  is 
directly  proportional  to  V. 

Unfortunately  there  seems  no  rational  formula  for  V.  "The 
formula  V2  =  u2  —  2  g(pi  —  p)  is  theoretically  correct,  but  there 
is  no  satisfactory  way  to  determine  or  fix  p  in  this  formula  pre- 
liminary to  the  design.  This  fact  necessitated  dependence  on 
the  cut-and-try  process  by  the  pioneers  in  this  field;  though  now 
data  have  been  gotten  for  so  many  pumps  and  fans  of  various 
styles,  showing  the  relation  between  head  and  discharge,  that 
designers  can  proceed  with  tolerable  confidence  except  where 
some  radical  departure  in  design  is  to  be  tried  (see  Art.  53). 

Assuming  that  the  designer  has  the  data  showing  relation 
between  H  and  Q  (or  V)  for  the  style  of  machine  he  is  going  to 
copy,  he  has  equations  Va  and  F&  to  conform  to. 

The  angle  ft  should  be  selected  with  due  regard  to  the  service 
required  of  a  machine  and  method  of  propulsion.  Notice  that, 
assuming  u  constant,  when : 

Angle  ft  is  less  than  90°,  H  increases  as  Q  increases. 

Angle  ft  is  equal  to  90°,  H  is  independent  of  Q. 

Angle  ft  is  greater  than  90°,  H  decreases  as  Q  increases. 

It  is  common  to  assume  the  fluid  as  approaching  the  inner 
limits  of  the  fan  blades  in  a  radial  direction  (see  Fig.  19)  even 
when  no  guide  vanes  are  provided,  though  in  that  case  the  as- 
sumption may  be  far  from  the  truth. 

Note  also  that  with  H  fixed,  u  must  increase  as  ft  increases. 
This  fact  is  taken  into  consideration  when  a  machine  is  to  be 


FANS 


99 


driven  by  a  high-speed  electric  motor  or  a  steam  turbine.  In 
such  cases  the  embarrassing  condition  in  the  design  is  to  apply 
the  high  rotative  speed  without  getting  excessive  head;  hence, 
in  such  cases  the  angle  0  is  made  greater  than  90°. 

Another  advantage  in  turning  the  vanes  backward  (|8  greater 
than  90°)  in  case  of  electric-driven  machines,  is  that  the  machine 
will  not  be  overloaded  when  the  head  or  resistance  is  suddenly 
thrown  off,  with  the  resulting  great  increase  in  discharge  (which 
increases  V)  (see  Fig.  21). 

In  cases  where  a  machine  is  to  be  run  by  a  reciprocating 
steam  engine  direct-connected  and  the  designer  has  trouble  in 


12 
11 
m 

Pressure  in  ITiohes  of  Water 

~  _  ro  co  ,u.  tn  o  -4  oo  to  < 

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ead 

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££ 

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^-—  ' 

«^—  - 

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-903 

"—  —  — 

. 

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"H 

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V., 

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Xx 

34567 
Cubic  Feet  per  Minute -Thousands 

FIG.  21. — Characteristic  curves.    Constant  speed.    Varying  discharge. 

getting  the  desired  head  with  the  limited  speed,  he  will  find  it 
advantageous  to  turn  the  vanes  forward. 

Where  a  constant  head  (or  pressure)  is  desired  with  varying 
quantity  as  in  sewage  pumps  and  ventilating  fans  for  buildings, 
the  most  rational  design  would  be  to  provide  an  adjustable  dis- 
charge with  radial  vanes. 

Another  condition  that  should  receive  consideration  in  de- 
signing, or  selecting,  a  machine  is  where  a  pump  is  to  force  water 
through  a  long  pipe  and  where  a  fan  is  to  force  air  through  a  mine. 
In  such  cases  the  greater  portion  of  the  resistance  to  be  over- 
come is  friction  head,  and  it  is  well  known  that  this  varies  with 
the  square  of  the  velocity,  and,  therefore,  any  increase  in  the 


100  COMPRESSED  AIR 

quantity  will  be  accompanied  with  a  relatively  greater  resisting 
head.  This  case  would  be  best  met  by  setting  the  vanes  radial 

u2 
(at    discharge).     Then,    theoretically,    H  =  -      and    quantity 

o 

varies  directly  with  u,  when  running  under  most  favorable  con- 
ditions. Now,  as  stated  before,  friction  will  vary  as  quantity 
squared.  Hence,  H  will  vary  directly  with  the  friction.  This 
very  nearly  meets  the  most  desired  condition  in  mine  ventilation 
where  practically  all  the  resistance  is  due  to  friction.  To 
illustrate  numerically:  Suppose  it  is  decided  to  double  the 
quantity  of  air  passing  through  a  mine.  If  we  double  the  speed 
of  the  fan  we  get  double  the  flow  of  air,  four  times  the  pressure, 
four  times  the  friction  and  eight  times  the  power  will  be  neces- 
sary to  run  the  fan. 

Probably  the  engine  or  motor  in  the  above  example  would  be 
incapable  of  developing  eight  times  its  normal  power.  How 
then  can  the  problem  be  solved?  Will  it  be  effective  to  install 
a  duplicate  of  the  first  fan  and  discharge  both  into  the  mine? 
No,  for  we  would  be  trying  to  put  through  double  the  quantity 
without  increasing  the  pressure.  The  result  would  be  a  some- 
what greater  pressure  and  quantity,  but  both  fans  would  now 
be  working  inefficiently,  if  they  were  properly  adopted  to  the 
first  condition. 

Would  the  problem  be  solved  by  placing  another  fan  in  series 
(or  tandem)  with  the  first?  No,  for  now  we  would  be  proposing 
twice  the  head  with  no  increase  of  volume.  There  would  be  some 
increase  in  volume,  but  not  double,  and  again  the  fans  would 
not  be  working  under  best  conditions. 

By  this  simple  example  it  is  evident  that  if  there  is  a  radically 
different  quantity  to  be  sent  through  a  long  conduit  (pipe,  flume 
or  mine),  the  only  scientific  solution  is  to  install  a  new  machine 
adapted  to  work  efficiently  under  the  new  conditions. 

Art.  52.  Testing. — The  following  discussion  refers  to  fans  or 
blowers. 

Manufacturers  of  recognized  standing  have  facilities  for  testing 
their  machines  and  should  know,  with  sufficient  accuracy  for 
commercial  purposes,  what  their  machines  will  do  and  the  condi- 
tion under  which  they  will  work  most  efficiently,  and  the  pur- 
chaser of  a  machine  for  any  important  serivce  should  demand  a 
guaranteed  performance  chart  for  the  machine,  this  chart  to  give 
information  equivalent  to  that  shown  on  Fig.  22.  Then,  in  the 


FANS  101 

acceptance  test  the  engineer  for  the  purchaser  might  be  content 
with  checking  a  few  points  on  the  performance  chart  under  con- 
ditions approximating  those  under  which  the  machine  is  to  work. 

In  the  purchaser's  test,  the  data  wanted  are  quantity,  head  and 
efficiency.  Too  often  the  purchaser  is  content  with  determining 
the  first  two  (or  even  with  no  test  at  all  if  the  machine  runs  and 
does  some  work).  A  test  will  require  the  service  of  a  technical 
man,  but  under  competent  direction  should  not  be  difficult  nor 
expensive. 

The  head  will  be  measured  in  inches  of  water  in  a  U-tube  (see 
precautions,  Art.  23a).  The  quantity  may  be  determined  by 
measuring  velocity  and  area.  Where  very  great  quantities  are 
passing,  the  annamometer  is  the  most  convenient  instrument  for 
measuring  velocity,  but  it  should  not  be  depended  on  in  unskilled 
hands.  It  will  need  careful  rating  and  should  be  applied  in 
all  parts  of  the  cross-section  of  the  conduit,  and  the  total 
quantity  found  by  summing  the  products  of  small  areas  by 
their  respective  velocities.  In  doing  such  work  the  operator's 
confidence  in  the  method  is  apt  to  be  shaken  by  the  discovery 
that  the  velocity  will  vary  considerably  over  the  area  and  will 
also  vary  with  time  at  a  fixed  point.  In  any  case,  the  section 
at  which  the  velocity  is  taken  should  be  well  away  from  the 
machine,  else  the  irregular  currents  will  render  the  observations 
altogether  unreliable. 

The  author  is  partial  to  orifice  measurement,  even  for  testing 
the  largest  fans.  Orifice  coefficients  are  now  available  up  to  30 
in.  diameter  or  30  in.  square  (see  Art.  23).  It  is  the  author's 
opinion  that  a  coefficient  of  0.60  will  result  in  errors  well  within 
those  made  in  reading  water  gages,  and  quite  certainly  with 
errors  less  than  are  apt  to  enter  any  annamometer  or  petot-tube 
measurements.  Note  that  the  orifice  coefficient  is  constant, 
while  that  of  a  petot  tube  or  a  revolving  annamometer  must  be 
found  for  each  instrument  and  may  change  with  the  slighest 
injury  or  misuse  of  the  instrument,  and  note  further  that  with 
reasonable  care  that  cross-currents  are  not  allowed  in  front  of  the 
orifice,  its  discharge  is  not  effected  by  unequal  velocities  in  the 
cross-section  of  the  conduit. 

Omitting  any  discussion  of  apparatus  for  measuring  velocities, 
quantities  and  pressures,  with  their  calibration  and  defects,  the 
engineer  will  need  to  determine: 


102 


COMPRESSED  AIR 


v  = 

W  = 
P  = 

N  = 
Q  = 
EI  = 
E2  = 
Then, 


the  velocity  of  air  passing  the  section  of  area,  a  (feet  per 

second), 

weight  of  air  passing  (pounds  per  second), 

pressure  of  air  at  section  =  5.21  i  where  i  is  pressure  in 

inches  of  water  (pounds  per  square  feet), 

revolutions  per  minute,  u  =  2TrrN, 

volume  passing  =  av  (cubic  feet  per  second). 

power  put  into  the  fan  (foot-pounds  per  second), 

the  useful  work  done  by  the  fan. 


E< 


and  efficiency  =  ^r- 

JuS  1 

He  should  also  have  all  dimensions  and  angles  of  the  fan  in 
order  better  to  interpret  results. 


012345G789 
Cubic  Feet  per  Minute  in  Thousands 

FIG.  22. 

The  variables  are  N,  v,  and  P.  In  a  thorough  test  to  get  the 
performance  chart  of  a  fan,  the  preferable  method  is  to  maintain 
a  constant  N  throughout  a  series  of  runs  in  which  P  is  varied  at 
will  by  the  operator,  v  measured  and  E  computed. 

Then,  with  another  N  another  series  is  run  as  before  and  so 
covering  the  desired  range  for  the  fan.  From  these  notes  the 
performance  curves  can  be  drawn.  The  most  important  of  these 
are  those  for  efficiency  and  for  quantity  (see  Fig.  22) . 


FANS  103 

The  measures  of  v  and  P  should  be  taken  in  a  section  of  smooth 
straight  conduit  some  distance  from  the  fan.  The  pressure  is 
controlled  by  placing  some  kind  of  shutter  in  the  conduit  beyond 
the  section  at  which  v  and  P  are  measured. 

A  performance  chart  of  the  class  shown  in  Fig.  22  shows  in 
remarkable  completeness  what  can,  and  should,  be  accomplished 
by  the  machine  and  under  what  conditions  it  will  work  most 
efficiently.  On  this  chart  the  pressure  curves  and  efficiency 
curves  are  plotted  in  the  usual  way  as  suggested  above,  then  the 
efficiency  contours  are  located  thus.  To  locate  the  55  per  cent, 
contour,  find  the  two  points  where  the  1,000-r.p.m.  efficiency 
line  crosses  the  55  per  cent,  line  (at  about  the  5.2  and  7.7  ver- 
ticals). Shift  these  points  vertically  to  the  1,000-r.p.m.  pressure 
line  and  mark  55.  Similarly  find  where  the  800-r.p.m.  efficiency 
line  cuts  the  55  per  cent,  efficiency  line  (at  3.6  and  6.).  Shift 
these  up  (or  down)  to  the  800-r.p.m.  pressure  line  and  mark  55 
as  before,  etc.  Connect  the  points  of  equal  efficiency  by  a 
curve.  Similarly  the  60  per  cent,  contour  can  be  drawn.  Then 
evidently  the  best  combination  for  operating  the  machine  is 
within  the  area  surrounded  by  the  60  per  cent,  efficiency  con- 
tour. For  instance,  if  we  want  7,000  cu.  ft.  per  minute,  the 
machine  should  be  speeded  to  about  1,000  r.p.m.  and  at  these 
rates  the  pressure  would  be  about  7  in.  Of  if  we  want  5,000 
cu.  ft.  per  minute  the  speed  should  be  about  800  and  the  pressure 
about  5  in. 

Art.  53.  Suggestions. — The  following  suggestions  seem  to  be 
the  rational  conclusions  pointed  to  by  theory,  the  difficulties  in 
controlling  operation,  and  complications  in  analyzing  the  results 
of  tests. 

Observing  the  rules  as  to  smooth  curves,  polished  surfaces,  and 
correct  angles,  design  a  high-speed  fan  with  characteristics  as 
revealed  in  Fig.  23.  DBE  is  an  adjusting  tongue  hinged  at  D. 
By  this  area  AB  can  be  varied  at  will.  The  area  of  the  sections 
gh,  etc.,  are  so  proportioned  as  to  maintain  the  velocity  u  in  the 
volute  until  the  throat,  or  switchpoint,  A,  is  passed,  after  which 
the  velocity  is  gradually  reduced  and  pressure  increased  (by  the 
well-known  laws  of  fluid  dynamics)  in  the  trumpet-shaped  outlet. 

In  operation  the  intent  would  be  to  keep  a  constant  pressure  in 
the  volute  up  to  AB}  this  pressure  head  being  approximately 

uz 
~ '     Then  the  whole  theory  of  this  machine  would  be 


104 


COMPRESSED  AIR 


f\  TT  J 

Q  =  au,  a.  —  —  and 


WCLU3 

- 


A  factory  test  of  such  a  machine  would  reveal  the  most  favor- 
able relation  between  u  and  p.  Then  a  size  would  be  selected 
that  would  give  the  desired  Q.  In  operation  there  would  be  a 


Sliding  Contact 


FIG.  23. 

water  gage  tapped  into  the  section  AB  to  show  p  at  any  time. 
When  the  operator  notes  that  p  is  low,  he  will  open  up  the  area  at 
AB  and  vice  versa. 

Note  particularly  that  in  such  a  design  Q  can  be  controlled 
independently  of  H. 


CHAPTER  IX 
CENTRIFUGAL  OR  TURBO  AIR  COMPRESSORS 

Art.  54.  Centrifugal  Compression  of  an  Elastic  Fluid. — The 

demonstrations  given  in  the  preceding  chapter  apply  to  any  case 
where  there  is  no  change  of  density  of  the  fluid  while  passing 
through  the  machine,  and  this  includes  the  case  of  centrifugal 
acceleration  without  change  of  pressure,  and  purely  impulsive 
action. 

We  have  now  to  study  the  case  where  compression  due  to 
centrifugal  force  within  the  wheel,  or  runner,  is  so  great  that  it 
must  be  considered  in  the  formulas. 

We  will  assume  isothermal  conditions,  since  the  ratio  of  com- 
pression in  each  stage  is  low  and  intercooling  can  be  applied 
between  each  stage.  The  formulas  thus  gotten  are  simpler  than 
can  be  gotten  otherwise,  and  are  as  accurate  as  is  justified  by 
other  considerations. 

In  Fig.  19  assume  the  cylinder  CB  filled  with  a  compressible 
fluid,  as  air.  The  weight  of  a  unit  volume  will  not  be  constant, 
but  will  depend  on  the  distance  x  from  the  center  and  on  the 
velocity  of  rotation. 

Let  wx  be  the  weight  of  a  unit  volume  at  distance  x  from  the 
center.  Then  the  centrifugal  force  due  to  a  disk  of  unit  area  and 
radial  thickness  dx  will  be 


Also 


wxux2  ,        wxu2    ,     .  x 

-dx  =  — -  xdx  since  ux  =  - 

gx  gr2  r 


where  px  is  absolute  pressure  in  the  air  at  distance  x  from  the 
center,  and  Wi  and  p1  are  the  weight  and  pressure  respectively 
of  the  air  at  entrance. 

Substituting  and  dividing  by  px  there  results 


,  then  I    ^  =  «!«!  I   «to. 

Jvi     x          ir2Jn 


_•  =  xdx, 

Px       Pigr2  vi  px       pigr 

Whence 

105 


106  COMPRESSED  AIR 


p         WM2    /„         9\       Wi/u2-ui2\     .  r 

—  =  —  -  —  -(r2  —  ri2)  =  —  (  —  =  --  )  since  n  =  -u\. 
Pi      Pi'2gr2\  /        pi\     2g     I  u 

If  ri  =  0  and  we  consider  a  single-stage  machine  taking  in  free 
air  we  will  have 


where  Rf  is  the  ratio  of  compression  at  the  periphery  but  within 
the  revolving  wheel. 

Assume  that  the  machine  is  in  1  second  putting  a  volume  of  free 
air  =  va  into  the  state  of  pressure  and  motion  indicated  above. 
Then  the  work  done  per  second  will  be 


u2  u2 

R'  +  WaVa  —  =  WaVa  —  (6) 

y  y 

when  the  value  of  logc  R'  from  (a)  is  inserted. 

Note  that  this  is  the  same  as  would  result  if  the  machine  were 
working  on  an  inelastic  fluid  of  weight  wa  (see  Art.  50). 

Note  also  that  the  work  done  in  compression  is  equal  to  that 
done  in  giving  velocity. 

Art.  55.  A  More  Direct  Derivation  of  Equation  (37).  —  Appli- 
cable also  to  Centrifugal  Air  Compressors  : 

After  a  study  of  Arts.  49,  50  and  54  the  student  will  be  pre- 
pared for  the  following  more  general  and  more  direct  demonstra- 
tion of  Eq.  (37)  and  its  application  in  case  of  considerable  com- 
pression. 

Referring  to  Figs.  24  and  20.  The  static  pressure  in  the  fluid 
changes  as  it  passes  out  of  the  rotating  part  into  the  fixed  out- 
let passage.  It  is  this  drop  in  pressure  that  induces  the  relative 
discharge  velocity,  V.  This  difference  in  pressure  offers  no 
resistance  to  the  rotation  of  the  wheel;  as  will  be  readily  seen  if 
we  imagine  the  perifery  of  the  wheel  closed  while  rotating  in  a 
frictionless  fluid.  The  pressure  in  the  frictionless  fluid  must  be 
normal  to  the  perifery  and  therefore  does  not  resist  its  rotation. 

Then  in  all  cases  (regardless  of  change  of  pressure  at  outlet) 
the  resistance  to  rotation  is  due  solely  to  the  reaction  of  the 
departing  jet.  This  reaction  is  in  direction  opposite  to  that  of 
the  absolute  velocity  of  discharge,  v,  (Fig.  19)  and  in  amount 

is  TF-.    But  the  component  opposed  to  rotation  (that  is  in  direc- 

tion opposite  to  u)  is  W    cos  6  and  as  is  apparent  on  the  dia- 

y 


CENTRIFUGAL  OR  TURBO  AIR  COMPRESSORS          107 

grams  v  cos  6  =  u  +  V  cos  /?.     Therefore  the  force  opposed  to 

W 

rotation  is  —  (u  +  V  cos  ft  and  since  work  done  by  the  wheel 

y 
equals  force  multiplied  by  distance.     Then 

Work  =    -  (u2  +  uV  cos  ft. 

Evidently  this  is  independent  of  the  radial  depth  (r  —  ri)  of  the 
vanes.  Then  the  radial  depth  of  vanes  is  a  matter  of  convenience 
or  expediency. 

In  case  of  a  fluid  of  uniform  density  (in  low  pressure  fans  we 
may  neglect  change  of  density)  if  the  machine  imparts  a  head, 
H,  to  the  fluid,  then  work  =  WH  :  Whence 


WE  =  —  (u2  +  uV  cos  ft 


and 


H  =  -      -^^  (37) 

In  case  of  a  compressible  fluid  (as  air)  and  we  are  to  consider 
the  work  done  in  compression. 

Let  RI  be  the  final  ratio  of  compression  when  the  air  has  been 
brought  to  rest  after  one  stage.  Then  work  =  pava  logc  R\ 
where  va  is  the  volume  of  free  air  compressed.  Then 

PaVa  logc   Rl    =   — —  (U2  +  UV  COS  ft 

and 

'u2  +  u>V  cos 


V  COS  p\  .         . 

-  --  1  (37a) 


This  is  the  formula  for  ratio  of  compression  produced  by  one 
stage  of  a  centrifugal  air  compressor. 

All  the  discussion  in  Art.  51  concerning  the  effects  of  the  angle 
j3  applies  also  to  equation  (37a).  The  student  should  read  that 
article  as  a  part  of  this  study. 

If  there  are  n  stages  in  a  machine,  each  giving  an  additional 
ratio,  Ri,  and  the  final  ratio  from  free  air  be  Rn,  then 

Rn  =  Rin  and  logc  Rn  =  n  logc  R!  (38) 

Art.  56.  Working  Formula.  —  The  very  great  centrifugal  force 
developed  in  these  machines  prompts  manufacturers  generally  to 
prefer  to  set  the  outer  tips  of  the  propeller  blades  radial  (0  =  90°) 


108  COMPRESSED  AIR 

to  avoid  cross  bending.     This  is  good  practice  for  other  reasons 
(see  Art.  51)  one  being  that  formulas  for  designing  and  analysis 
are  much  simplified,  as  the  following  will  show: 
In  Eq.  (37a)  assume  /?  =  90°.     Then 

wau2 
logc  fli  =  — -' 

Pa    Q 

Note  that  w  =  K0  0-  .  and,  if  t  be  assumed  constant,  — -  will 
OO.OD  t  pag 

be  constant.    To  adopt  the  formula  to  common  logarithms  (which 
will  be  more  convenient)  divided  by  2.3026. 

In  such  a  machine  perfect  cooling  cannot  be  accomplished. 
We  will  assume  for  this  study  an  average  temperature  of  580 
(120°F.).  Then  the  formula  becomes 

loglo  fii  =  (^3026  X  53.35^  580  X  32.2)  u*  =  *"'       (39) 
log  k  =  7.6398. 

In  the  following  examples  (taken  from  practice)  (3  =  90°. 

Example  1. — A  three-stage  machine,  54  in.  in  diameter,  r.p.m. 
2,500,  in  operation,  gives  15  Ib.  gage  pressure  and  delivers  35,000 
cu.  ft.  per  minute  of  free  air,  the  power  necessary  being  2,700  hp. 

Determine  the  efficiency  of  the  machine  as  to  pressure  and 

power. 

9  ^00  ^4 

Here  u  =££     X  3.14  X  ?!  and    log  w2  =  5.5368  u  590 

OU  1.^5 

log  k  =  7.6398 
log  of  log  R!  =  T.1766 

logfii  =  0.1500        Ri    1.40 
3 

log  Rn  =  0.4500        Rn    2.83 

Assuming  pa  =  14.4  where  the  machine  is  in  operation, 
then  p  =  2.83  X  14.4  =  40.75  and  gage  pressure  =  40.75  - 
14.4  =  26.3. 

Then  efficiency  as  to  gage  pressure  ^^  =  57  per  cent,  and 
theoretic  efficiency  as  to  work  would  be,  by  Eq.  32, 
log  R       log  2.04      0.3096 
lofe  =  Iok83  =  04518  =  68  P 

The  report  of  the  test  of  the  machine  gave  the  "shaft"  effi- 
ciency as  71  per  cent.,  the  meaning  not  being  further  defined. 


CENTRIFUGAL  OR  TURBO  AIR  COMPRESSORS          109 

Example  2.  —  A  single-stage  machine,  34  in.  in  diameter  with 
3,450  r.p.m.  gave  3.25  gage  pressure  and  the  horsepower  was  350 
for  18,000  cu.  ft.  per  minute. 

What  efficiency  as  to  pressure  and  power  did  the  machine  show? 


u  =  X  3.14  X  j    and  log  u2=  5.4048 

log  k  =  7.6395 
log  of  log  #1  =  T.0443 
log#i  =  0.1107 
fli  =  1.29 

Assuming  14.5,  then  P  =  (1.29  -  1)  14.5  =  4.2  nearly. 
The  ratio  efficiency  would  be  -  ~fj~5~  "  =  (1-22)  divided  by 

3  25 
1.29  =  95  per  cent,  and  gage  pressure  efficiency  =  -j-^-  =  77 

per  cent. 

Horsepower  necessary  to  compress  18;000  cu.  ft.  per  second 
to  R  =  1.22  is  218.     Therefore,  the  efficiency  as  to  power  = 
218 
350  =63  per  cent. 

Example  3.  —  A  six-stage  machine  27  in.  in  diameter,  r.p.m. 
3,450,  gives  15  Ib.  gage  and  340  hp.,  capacity  4,500  cu.  ft.  per 
minute. 

o  4.Kfj  07 

u  =  ^  X  3.14  X  ^  logu2  =  5.1976 

log  k  =  7.6395 
log  of  log  Ri  =  2.8371 

logfli  =  0.0687  Ri  =  1.17 

_  6 
log  Rn  =  0.4122 

Rn  =  2.58,  Pn  =  37.5 
Pg  =  23  Ib. 

The  ratio  accomplished  by  the  machine  is  2  very  nearly;  there- 

2 
fore,  the  ratio  efficiency  ^-^  =  77  per  cent. 

Z.Oo 

The  work  necessary  to  compress  4,500  cu.  ft.  per  second  to 
R  =  2  is  198.  Therefore,  the  work  efficiency  is  58  per  cent. 

When  pressure  is  low,  as  in  Ex.  2,  the  estimated  efficiencies  will 
be  materially  effected  by  the  atmospheric  pressure  and  the  pres- 


110 


COMPRESSED  AIR 


sure  developed  should  be  determined  by  a  water  or  mercury 
column;  otherwise  only  rough  approximations  will  be  obtained. 

Art.  57.  Suggestions. — The  following  considerations  point  to 
the  conclusion  that  best  results  will  be  gotten  from  centrifugal  air 
compressors  when  the  air  is  held  in  the  machine  until  every  par- 
ticle is  under  full  centrifugal  pressure  regardless  of  its  position 
relative  to  the  propellers.  Then  it  will  escape  through  the  outlet 
passages  with  uniform  velocity  and  pressure,  a  condition  evi- 
dently essential  to  high  efficiency.  Otherwise,  if  the  air  is  still 
under  the  impulsive  pressure  of  the  vanes  as  it  escapes  from  the 
machine,  those  particles  next  the  propeller  as  at  d,  Fig.  24,  must 
be  under  greater  pressure  than  those  at  c,  and  the  velocity  of 


FIG.  24. 

escape  relative  to  the  revolving  machine  will  be  greater  at  d  than 
at  c. 

In  these  machines  the  velocity  of  rotation,  u,  is  always  very 
high  and  any  moderate  relative  velocity  of  discharge  (say  within 
100  ft.  per  second)  will  leave  the  absolute  path  of  the  escaping  air 
nearly  on  a  tangent  to  the  perimeter,  as  at  ab.  This  being  the 
case,  a  flaring  fixed  receiving  passage  about  as  shown  at  (6), 
Fig.  24,  would  cause  the  velocity  to  be  gradually  checked.  It 
is  not  apparent  that  any  advantage  will  be  gotten  by  putting 
tongues  ef  in  this  outlet  passage.  They  would  increase  friction 
without  apparent  compensating  benefit. 

Note  that  a  section  of  the  flaring  outlet  on  a  horizontal  plane 
through  ab  will  show  a  much  longer  path  than  the  radial  section 
(see  (c),  Fig.  24). 

The  very  great  centrifugal  stress  in  these  machines  lead  manu- 


CENTRIFUGAL  OR  TURBO  AIR  COMPRESSORS          111 

facturers  generally  to  prefer  to  set  the  outer  end  of  the  propellers 
radial,  and  this  is  good  practice  for  other  reasons,  one  being  the 
simplified  formula  for  designing. 
Art.  58.  Proportioning. — 
Let      va  =  volume  of  free  air  to  be  compressed,  cubic  feet 

per  second, 

r  =  radius  to  outlet  of  propeller, 
n  =  radius  to  inlet  of  propeller, 
u  =  velocity  of  rotation  at  outlet, 
HI  =  velocity  of  rotation  at  inlet, 

5  =  radial  component  of  the  velocity  at  outlet, 

Si  =  radial  component  of  velocity  of  air  entering  at 

radius  n, 
0  =  angle  between  forward  direction  of  ui,  and  tangent 

to  vane  at  inlet, 

6  =  width  of  outlet, 
61  =  width  of  inlet. 

All  linear  units  in  feet. 

R'  =  ratio  of  compression  of  air  within  the  wheel  at  the  outlet 

but  before  escaping, 
Ri  =  ratio  of  compression  when  brought  to  rest  at  end  of  first 

stage. 
Then 

tan  <£  =  - 

Ui 

and  Va  =  2  vrJ>iSi. 

Usually  61  is  to  be  determined  by  this  relation,  all  other  factors 
being  known. 

Note  that  this  equation  holds  only  when  Si  is  the  radial  com- 
ponent of  outward  movement  into  the  vanes.  When  there  are 
no  guide  vanes  at  entrance,  Si  becomes  uncertain  and  erratic. 
When  it  becomes  the  practice  to  put  guide  vanes  at  entrance, 
much  of  the  uncertainties  in  the  design  of  such  machines  will  be 
removed. 

If  there  are  to  be  n  stages  and  a  final  ratio  of  compression 

i 

Rn,  then  RI  =  Rn«  and  R'  =  RiH.    u  will  be  fixed  by  the  re- 
lation from  (38) 


112  COMPRESSED  AIR 

When  u  is  determined,  r  and  r1  can  be  assigned  between  limits 
found  advisable  by  experience  and  the  necessity  of  having  pas- 
sages of  sufficient  area. 

At  the  outlet  the  width  b  is  fixed  by  the  relation 


The  greatest  difficulty  in  the  theoretic  design  of  this  class  of 
machines  is  in  correctly  predicting  the  factor  S  (or  relative  ve- 
locity of  discharge)  .  It  is,  of  course,  quite  sensitive  to  changes  of 
pressure  in  the  discharge  ducts.  It  is  this  doubtful  factor 
chiefly  that  forces  the  designer  to  depend  on  results  of  tests. 
Fortunately,  the  width  can  be  varied  without  affecting  any 
other  factors  except  Va>  Hence,  after  test  of  a  design  of  machine 
the  desired  capacity  can  be  gotten  by  varying  b  and  61. 

The  discharge  of  a  centrifugal  blower  can  be  made  adjust- 
able without  varying  the  pressure  by  the  simple  device  shown  in 
Fig.  23. 

Finally,  the  student  should  be  reminded  that  the  above 
mathematical  formulas  do  not  include  losses  due  to  friction  nor 
imperfections  of  design.  Their  chief  value  is  to  show  what 
would  be  realized  in  a  perfect  machine  and  so  reveal  the  short- 
comings of  a  machine  and  guide  the  designer  in  modification  for 
improvements. 


CHAPTER  X 
ROTARY  BLOWERS 

Art.  59. — In  certain  lines  of  manufacture,  there  is  necessary  a 
supply  of  air  in  great  volume,  and  at  pressures  not  found  prac- 
ticable for  fans  and  yet  so  low,  that  to  build  reciprocating  com- 
pressors to  meet  the  demand  would  seem  extravagant,  when  the 
cost  is  compared  to  the  power  demanded. 


FIG.  25. 

This  demand  has,  in  the  past,  been  most  economically  met  by 
the  class  of  machines  known  as  rotary  blowers.  These  vary  in 
details  and  there  are  several  patterns  on  the  market.  Perhaps 
the  simplest  and  best  known  is  illustrated  in  Fig.  25.  The  two 
propellers  revolve  as  shown  by  the  arrows,  and  as  is  apparent  by 
inspection  the  pockets  of  fluid  (air  or  water)  are  forced  upward. 
The  flow  is  continuous,  but  not  uniform ;  neither  is  the  tort  on  the 
shafts  constant.  The  irregular  discharge  and  tort  tend  to  cause 

8  113 


114  COMPRESSED  AIR 

vibrations  but  this  is  met  by  making  the  machines  heavy  and 
rigid  (and  in  case  of  pumps  by  putting  air  chambers  near  both  in- 
let and  outlet).  There  are  no  valves  in  the  machine,  and  the 
makers  do  not  design  them  to  rub  or  move  in  contact  at  any  sur- 
face within  the  casing,  but  depend  on  accurate  workmanship  to 
make  the  clearance  between  the  surfaces  so  small  as  to  render  the 
leakage  so  small  as  to  be  tolerable. 

Then,  having  no  valves  nor  rubbing  surfaces,  the  machines 
handling  air  should  be  quite  durable,  but  this  cannot  be  said  of 
them,  when  used  to  pump  water  containing  any  grit.  It  is  a  good 
practice  to  supply  a  liberal  quantity  of  thick  oil  to  a  blower,  not 
for  lubrication,  but  to  reduce  clearance  between  the  surfaces. 

It  is  necessary  to  note  one  important  difference  in  the  working 
of  this  class  of  machine  as  a  blower,  and  as  a  pump.  This  is  due 
to  the  reexpansion  of  the  compressed  air  out  of  chamber  B  into  A, 
as  soon  as  communication  is  opened  between  the  two  chambers. 
This  is  lost  work  and  would  limit  the  pressure  at  which  the  ma- 
chine could  operate  economically,  even  if  slippage  did  not  increase 
with  pressure.  For  these  reasons  the  useful  range  of  pressures 
on  such  machines  seems  to  be  between  J^  and  5  Ib.  For  pres- 
sures below  H  Ib-  fans  are  usually  selected,  on  account  of  the  less 
cost.  For  pressures  above  5  Ib.  reciprocating  compressors  are 
usually  selected,  on  account  of  the  better  efficiency. 

Apropos  to  this  phase  of  the  subject,  read  Chapter  IX  on 
" Centrifugal  Air  Compressors." 


CHAPTER  XI 
EXAMPLES  AND  EXERCISES 

Art.  60. — The  following  combined  example  includes  a  solution 
of  many  of  the  types  of  problems  that  arise  in  designing  com- 
pressed-air plants.  The  student  will  find  it  well  worth  while  to 
become  familiar  with  every  step  and  detail  of  the  solutions,  which 
are  given  more  fully  than  would  be  necessary  except  for  a  first 
exercise. 

Example  60. — An  air-compressor  plant  is  to  be  installed  to 
operate  a  mine  pump  under  the  following  specifications : 

1.  Volume  of  water  =  1,500  gal.  per  minute. 

2.  Net  water  lift  =  430  ft. 

3.  Length  of  water  pipe  =  1,280  ft. 

4.  Diameter  of  water  pipe  =  10  in. 

5.  Length  of  air  pipe  =  1,160  ft. 

6.  Atmospheric  pressure  =  14  Ib.  per  square  inch. 

7.  Atmospheric  temperature  50°F. 

8.  Loss  in  transmission  through  air  line  =  8  per  cent,  of  the 
pv  logs  T  at  compressor. 

9.  Mechanical  efficiency  of  the  pump  =  90  per  cent,  as  re- 
vealed by  the  indicators  on  the  air  end  and  the  known  work 
delivered  to  the  water. 

10.  Average  piston  speed  of  pump  =  200  ft.  per  minute. 

11.  Mechanical  efficiency  of  the  air  compressor  =  85  per  cent. 
as  revealed  by  the  indicator  cards. 

12.  Revolutions  per  minute  of  air  compressor  =  90  and  vol- 
umetric efficiency  =  82  per  cent. 

13.  In  compression  and  expansion  n  =  1.25. 

Preliminary  to  the  study  of  the  problems  involving  the  air  we 
must  determine: 

(a)  Total  pressure  head  against  which  the  pump  must  work.  By 
the  methods  taught  in  hydraulics  the  friction  head  in  a  pipe  10 
in.  in  diameter,  1,280  ft.  long,  delivering  1,500  gal.  per  minute,  is 
about  20  ft.  Therefore,  the  total  head  =  450  ft. 

115 


116  COMPRESSED  AIR 

(b)  Total  work  (Wi)  delivered  to  the  water  in  1  min. 
Wi  =  1,500  X  S>£  X  450  =  5,625,000  ft.-lb. 

(c)  Total  work  (W)  required  in  air  end  of  pump.     By  specifica- 

tion 9,  W  =  ~~  =  6,250,000  ft.-lb.  =  190  hp. 

For  the  purpose  of  comparison,  two  air  plants  will  be  designed; 
the  first,  designated  (d)  as  follows: 

(d)  Compression  single-stage  to  80  Ib.  gage.     No  reheating. 
No  expansion  in  air  end  of  pump.     Pump  direct-acting  without 
flywheels. 

Determine  the  following: 

(dl)  Air  pressure  at  pump  and  pressure  lost  in  air  pipe.     By 
specification  8  and  Eq.  (32), 

log  ^ 


Whence,  using  common  logs,  log  j^  =  0.76118  and 

p2  =  80.78. 

Then  lost  pressure  =  pi  -  pz  =  94  -  80.78  =  13.22  =  f, 
and  gage  pressure  at  pump  =  80  —  13.22  =  66.78. 

(d2)  Ratio  between  areas  of  air  and  water  cylinders  in  pump. 
The  pressure  due  to  450  ft.  head  =  450  X  0.434  =  194.3,  say 
195  Ib.,  per  square  inch;  and  since  pressure  by  area  must  be 

,       area  air  end  195 

equal  on  the  two  ends,  -  —  ?  =  aa  no  =  3  nearly. 

area  water  end       66.78 

(c?3)  Volume  of  compressed  air  used  in  the  pump.  Cubic  feet 
per  minute.  Evidently  from  solution  (d2)  the  volume  of  com- 
pressed air  used  in  the  pump  will  be  three  times  that  of  the 
water  pumped,  or 


v  =  *jZj~  x  3  =  601.6  cu.  ft.  per  minute. 

(d4)  Diameters  of  air  cylinder  and  of  water  cylinder.  Since  the 
piston  speed  is  limited  to  200  ft.  per  minute  (specification  10)  and 
the  volume  is  1,500  gal.,  we  have,  when  all  is  reduced  to  inch  units 
and  letting  a  =  area  of  water  cylinder,  a  X  200  X  12  =  1,500 
X  231.  Whence  a  —  144  sq.  in.  which  requires  a  diameter  of 
about  13^  in. 


EXAMPLES  AND  EXERCISES  117 

The  area  of  air  cylinder  is  by  (d2)  three  times  that  of  the  water 
cylinder,  which  gives  a  diameter  23}^  in.  for  the  air  cylinder. 

(d5)   Volume  of  free  air.     From  (dl)  r  at  the  pump  =  5.76. 
Therefore 

va  =  601.6  X  5.76  =  3,465  cu.  ft.  per  minute. 

(dQ)  Diameter  of  <iir  pipe.     The  mean  r  in  the  air  pipe  is 
5.76  +  6.72 


get  d  =  5  in. 

13  22 
Or   using   Plate    III   with   r  X  13.22  +  1.160   or   r  X 


on  the/r  line  and  3,465  on  the  Va  line,  the  intersection  falls  near 
the  5-in.  point  on  the  d  line. 

(dl)  Horsepower  required  in  steam  end  of  compressor.  By 
Table  II  the  weight  per  foot  of  free  air  is  0.07422  Ib.  per  cubic  foot. 
Total  weight  of  air  compressed  =  Q. 

Q  =  0.07422  X  3,465  =  257  Ib.  per  minute. 

In  Table  I  opposite  r  =  6.72  in  column  9  find  by  interpolation 
0.3736.  Then 

Horsepower  =  2.57  X  0.3736  X  (460  +  50)  =  489.6  in  air 

,  489.6 
end,  and  A  0.    =  576  in  steam  end. 

U.oO 

The  second  plant  will  be  designated  by  the  letter  (e)  and  will  be 
two-stage  compression  to  200  Ib.  gage  at  air  compressor,  will  be 
reheated  to  300°  at  the  pump  and  used  expansively  in  the  pump; 
the  expansion  to  be  such  that  the  temperature  will  be  32°  at  end 
of  strtfke. 

(el)  Air  pressure  at  pump.  Apply  Eq.  (32)  as  in  (dl).  In  this 
case  7*1  (at  the  compressor)  ==  15.3  and  r2  (at  the  pump)  =  12.3. 
Therefore  pressure  at  the  pump  =  12.3  X  14  =  172.3  and  the 
lost  pressure  =  214  -  172.3  =  41.7  =  f. 

(e2)  Point  of  cutoff  in  air  end  of  pump  =  fraction  of  stroke 

during  which  air  is  admitted.     By  Eq.  (12),  viz..  ^  =  (—  )      , 

tl  W2/ 

putting  in  numbers  we  get  ^™  =  (  —  j       whence  —  =  0.176,  which 

is  the  point  of  cutoff,  and  v2  =  5.68  vi. 

760 
Or  go  into  Table  I  in  column  5,  find  the  ratio  -r^  =  1.545, 

and  in  same  horizontal  line  in  column  3  find  0.176. 


118  COMPRESSED  AIR 

(e3)  Volume  of  compressed  hot  air  admitted  to  air  end  of  pump. 
Apply  Eq.  (9),  viz.,  work  =  PlV^  _  ^2  +  piVi  -  pav2. 

In  this  we  have  work  =  6,250,00,  v2  =  5.68  vi,  pi  =  214, 
n  —  1  =  0.25,  pa  =  14,  and  p2  must  be  found  by  Eq.  (12a),  or 
it  may  be  gotten  from  Table  I  by  noting  that  for  a  tempera- 
ture ratio  of  1.545  the  pressure  ratio  is  8.8  and  -  =  0.1136. 

Therefore  p2  =  0.1136  X  172.3  =  19.57.     This  would  give  gage 
pressure  =  5.57. 

Inserting  these  numerals  in  Eq.  (9)  we  get 

6,250,000  =  144^172-3- 5-68X19.57  +  172.3  -  14  X  5.68) . 

Whence  v\  —  128.6  cu.  ft.  per  minute. 

(c4)  Diameter  of  air  cylinder  of  pump  when  air  and  water 
pistons  are  direct-connected.  Since  expansion  ratio  is  5.68  (see 
(e2))  and  the  volume  before  cutoff  is  128.6,  the  total  piston  dis- 
placement is  128.6  X  5.68  =  730.8  cu.  ft.  per  minute.  When 
the  air  and  water  pistons  are  direct-connected  they  must 
travel  through  equal  distances,  therefore,  the  air  piston  travels 
through  200  ft.  per  minute  (specification  10).  Then  if  a  =  area 
of  piston  in  square  feet  we  have 

200  a  =  730.8    and     a  =  3.654  sq.  ft. 

By  Table  X  the  diameter  is  26  in.  nearly. 

(c5)  Volume  of  cool  compressed  air  used  by  pump,  cubic  feet 
per  minute.  By  (e3)  the  volume  of  hot  compressed  air  is  128.6, 
and  since  under  constant  pressure  volumes  are  proportional  to 
absolute  temperatures,  we  have 

v  510 


128.6       760* 


Whence  v  =  86.3  cu.  ft.  per  minute. 


(e6)  Volume  of  free  air  used.  From  (el)  the  ratio  of  compres- 
sion at  the  pump  is  12.3  and  from  (e5)  the  volume  of  cool  com- 
pressed air  is  86.3,  therefore,  the  volume  of  free  air  is  86.3  X  12.3 
=  1,061.6. 

(el)  Diameter  of  air  pipe.     The  r  for  Eq.  (27)  is 

12.3 +JM  =  13.8. 


EXAMPLES  AND  EXERCISES  119 

Applying  Eq.  (21)  with  coefficient  c  =  0.07  we  have 


(e8)  Horsepower  required  in  steam  end  of  compressor.  By  (d7) 
the  weight  per  cubic  foot  of  free  air  is  0.07422  and  by  (e6)  the 
volume  of  free  air  compressed  is  ],061.6  Therefore,  the  total 
weight  compressed  is  0.07422  X  1,061.6  =  78.8  Ib.  per  minute, 
and  the  initial  absolute  temperature  is  510. 

In  the  two-stage  compression  r2  =  15.3,  and  assuming  equal 
work  in  the  two  stages  the  n  =  \/15.3  =  3.91  nearly  (see 
Art.  13).  Then  going  into  Table  I  with  r  =  3.91  in  column  9  find 
0.2525.  Hence  horsepower  =  0.2525  X  78.8  X  510  =  101.5  for 
one  stage,  and  for  the  two  stages  101.5  X  2  =  203, 

203 
and  (specification  11)  ^-^,  =  238.8  hp.  in  steam  end. 

U.oO 

(e9)  Diameter  of  air  compressor  cylinders,  assuming  3-ft.  strokes 
and  2%-in.  piston  rods,  equal  work  in  the  two  cylinders  and  allowing 
for  volumetric  efficiency.  By  (e6)  the  free  air  volume  is  1,061.6 
and  (specification  12)  the  volumetric  efficiency  =  82  per  cent. 
Therefore, 

1   OA1   A 

the  piston  displacement  =    '       '     =  1,294.6  cu.  ft.  per  minute. 

(J.OA 

By  specification  12  the  r.p.m.  =  90.  Therefore,  the  displace- 
ment per  revolution  =  14.7  nearly,  for  the  low-pressure  cylinder. 
Add  to  this  the  volume  of  one  piston  rod  length  of  3  ft.  which  is 
3X0  0.341  =  0.1023.  Whence  the  volume  per  revolution  must 

7.4 

be   14.8  or,   for  one  stroke,   7.4.     Whence  the  area  =  -5-   = 

o 

2.466  sq.  ft.     By  Table  X  the  diameter  is  21J4  in.  nearly  for  low- 
pressure  cylinders. 

The  high-pressure  cylinder  must  take  in  the  net  volume  of  air 
compressed  to  r  =  3.91  (see  (e8)).  Therefore,  the  net  volume  per 

1   Ofil   R 

revolution  =  nn'       'ni   =  3.02.     Add   one   piston  rod  volume 
yu  /\  o.yi 

and  get  3.12  per  revolution  or  1.56  per  stroke  and  an  area  of  0.53 
sq.  ft.     By  Table  X  this  requires  a  diameter  of  10  in.  nearly. 

(elO)  Temperature  of  air  at  end  of  each  compression  stroke.  In 
Table  I  the  ratio  of  temperatures  for  r  =  3.91  is  1.313.  Hence  the 
higher  temperature  =  510  X  1.313  =  669  absolute  =  209°F. 


120 


COMPRESSED  AIR 


DESIGN  OF  A  SYSTEM  OF  DISPLACEMENT  PUMPS 

The  water  in  a  mine  is  to  be  collected  by  a  system  of  displace- 
ment pumps,  one  each  at  B,  C,  D  and  E,  delivering  into  a  sump 
at  A. 

The  data  are  shown  on  the  sketch  (Fig.  26)  and  include: 
lengths  of  pipes  (1)  ;  elevations  (El)  and  quantities  (Q)  of  water 
in  cubic  feet  per  minute. 

The  lengths  of  water  pipes  and  air  pipes  will  be  assumed  equal. 
The  pipes  may  change  diameter  at  junctions.  Assume  one-third 
of  time  consumed  in  filling  the  tanks  with  water.  Then  the 
maximum  rate  of  discharge  must  be  three-halves  of  the  average. 

The  problem  is  to  specify  the  free  air  volume  (Va)  for  the  com- 
pressor and  the  gage  pressure  (P)  of  delivery.  Also,  the  diame- 
ters of  all  pipes  both  for  water  and  for  air. 


_  El=35 

D    Q=  8  El= 

Q=20E 


FIG.  26. 


Solution. — In  order  to  avoid  putting  in  reducer  valves  and  for 
economy  in  piping  we  will  as  nearly  as  practicable,  design  the 
system  so  that  static  head  +  friction  head  in  air  pipe  +  fric- 
tion head  in  water  pipe  shall  be  the  same  for  each  unit.  Evi- 
dently this  sum  will  be  fixed  by  conditions  at  E,  since  it  has 
greatest  lift  and  greatest  length  of  pipe. 

We  will,  therefore,  first  fix  diameters  for  lines  EH  and  HA, 
giving  them  liberal  dimensions  in  order  to  keep  down  the  pressure 
at  A  for  we  will  find  that  some  of  the  pressure  at  A  must  be 
wasted  when  working  the  pumps  at  B,  C  and  D. 

The  following  computations  were  made  with  the  aid  of  slide 
rule  and  logarithmic  friction  charts  (Plate  III),  such  method 
being  sufficiently  accurate  for  the  purpose. 


EXAMPLES  AND  EXERCISES 


121 


Water  line  EH : 
6  in.  diameter 

Water  line  HA : 
6  in.  diameter 


Length  1,400  ft.,  Q  =  %  X  20    =30 
Pressure  loss  in  1,400  ft. 
(pounds  per  square  inch  4.0 

Length  800,  Q  =42 

Pressure  loss  in  800  ft.  =  3.8 


Air  pipe  EH: 
2  in.  diameter 

Air  pipe  HA : 
2  in.  diameter 


Air  pipe  A  to  Compressor 
2  in.  diameter 


Water  friction  E  to  A  7.8 

Static  pressure  E  to  A  13.0 

Pressure  on  water  at  E  20.8 

For  20.8  Ib.  gage  r  =  2.44,  but  at  A  the  air 
pressure  must  be  somewhat  greater.     Hence, 
we  may  assume  r  =  2.5  for  estimating  friction 
in  pipes  leading  from  A. 
Length  1,400  ft.  volume  of  compressed  air 
Vc  =  30 

Va  =  r  X  30  =  75,  air  friction  #  toH,f  =  2.3 
Length  =  800  ft.  Vc  =  42,  Va  =  105,  /  =  2A 
Air  friction  E  to  A  =  4.7 
Air  pressure  at  A  =  20.8  +  4.7  =  25.5 
r  at  A  =  2.78 

Length  500  ft.,  Vc  =  %(12  + 
10  +  8  +  20)  =  75 
Va  =  r  X  75  =  210,  /=    4.8 

At  compressor  P  =  30.3 
The  assumption  that  all  pumps 
will  discharge  simultaneously  is 
extreme.    Hence  a  compressor  of 
200  cu.  ft.  per  minute  and  gage 
pressure  =  30  Ib.  will  be  ample. 
Now  with  air  pressure  =  25.5  at  junc- 
tion A ,  we  have  to  design  the  air  and  water 
pipes  to  pumps  B,  C  and  D  so  as  to  about 
use  up  this  pressure. 

Air  pipe  GA:     1  Length  800  ft.,  Vc  =  33,  Va  =  2.5  Vc, 
in.  diameter  J  Va  =  82,  /  =  3.2 

t,   Vc  =  15,  Va  =  37,   /  =  2.4 


Length  800  ft.,    Vc  =  18,    Va  =  45,  /  =  5.4 

From  the  above  we  find  air  pressure  in 
tank  B  =  25.5  -  (3.2  +  2.4)   =  18.9 

tank  C  =  25.5  -  (3.2  +  5.4)  =  16.9 


Air  pipe  CG : 
in.  diameter 


122 


COMPRESSED  AIR 


Water  pipe  AG :  ) 
5  in.  diameter  I 


Water  pipes  BG 
in.  diameter 


Water  pipe  CG  : 
4  in.  diameter 


Air  pipe  DH : 
%  in.  diameter 


Water  pipe  DH : 
in.  diameter 


Length  =  800   ft.,  Q  =  33,  friction   loss 
(pounds)  =  6.1 

Static  pressure  at  B  (20  ft.)  =  8.7  and 
18.9  -  (8.7  +  6.1)  =  4.1 

for  water  friction  BG 
Length  =  200  ft.,  Q  =  15,  loss  of  pres- 
sure =  2.2 
Leaving  a  margin  of  1.9  Ib. 
Length  800  ft.,  Q  =  18,  static  pressure 
(15  ft.)  =  6.5  and  16.9  -  6.5  =  10.4  that 
can  be  lost  in  friction  in  the  water  pipe. 
A  4-in.  pipe  will  take  up  6.2  leaving  a 
margin  of  about  4  Ib.     This  is  the  nearest 
commercial  size  that  can  be  used. 
Length  =  100  ft.,  Vc  =  12,  Va=  2.5  X 
12  =  30,  /  =                                                 3.6 
Air  pressure  in  D  =  25.5  —  air  friction  in 
AH  and  HD  =  25.5  -  {2.4  +  3.6)  =       19.5 
Static  water  pressure  at  D  (25  ft.)  =          10.9 
Available  for  water  friction  =                      8.6 
Length  100  ft.,  Q  =   12,  loss  in  2>i-in. 
pipe  =                                                            5.9 
Nearest  commercial  size,       Margin           2.7 


EXERCISES 

In  the  following  exercises,  where  not  otherwise  specified,  atmospheric  con- 
ditions may  be  taken  as  T  —  60°F.  and  pa  =  14.7. 

The  article  of  the  text  on  which  the  solution  chiefly  depends  is  indicated 
thus  (  )  and  the  answer  thus  [  ]. 

1.  (a)  Assuming  isothermal  conditions,  how  many  revolutions  of  a  com- 
pressor 16-in.  stroke,  14-in.  diameter,  double-acting,  would  bring  the  pressure 
up  to  100  Ib.  gage  in  a  tank  4  ft.  diameter  by  12  ft.  length,  atmospheric  pres- 
sure =  14.5  per  square  inch?     (1)  [361]. 

(6)  What  would  be  the  horsepower  of  such  a  compressor  running  at  100 
r.p.m.?  (1)  [37.3]. 

(c)  What  would  be  the  horsepower  if  the  compression  were  adiabatic? 
(2)  [51.0]. 

(d)  What  weight  of  air  would  be  passed  per  minute  when  r.p.m.  =  100  and 
T  =  60°F.?     (8)  [21.4]. 

2.  The  air  end  of  a  pump  (operated  by  compressed  air)  is  20  in.  in  diameter 
by  30-in.  stroke,  r.p.m.  =  50,  cutoff  at  Y±  stroke,  free  air  pressure  =  14.0, 
Ta  =  60°,  compressed  air  delivered  at  75  Ib.  gage,  T  =  60°  and  n  =  1.41. 

(a)  Find  work  done  in  horsepower.     (3)  [70]. 


EXAMPLES  AND  EXERCISES  123 

(6)  Find  weight  handled  per  minute.     (8)  [56] 

(c)  Find  temperature  of  exhaust  (degrees  F.).     (7)  [  —  165]. 

3.  With  atmospheric  pressure,  pa  =  14.7,  and  Ta  =  50°,  under  perfect 
adiabatic  compression,  what  would  be  the  pressure  (gage)  and  temperature 
(F.)  when  air  is  compressed  to: 

(a)  y%  its  original  volume?  (7)  [210]. 

(6)  H  its  original  volume?  (7)  [435]. 

(c)  Y§  its  original  volume?  <7)  [603]. 

(d)  Y%  its  original  volume?  (7)  [737]. 

(e)  Ho  its  original  volume?     (7)  [852]. 

4.  With  Pa  =  14.1  and  Ta  =  60°  what  will  be  the  pressure  of  a  pound  of 
air  when  its  volume  =  3  cu.  ft.?     (8)  [51.4]. 

6.  What  would  be  the  theoretic  horsepower  to  compress  10  Ib.  of  air  per 
minute  from  pa  =  14.3  and  Ta  =  60°  to  90  Ib.  gage? 
(a)  Compression  isothermal.     (1)  [16.7]. 
(6)  Compression  adiabatic.     (2)  [22.7]. 

6.  Find  the  point  of  cutoff  when  air  is  admitted  to  a  motor  at  250°F.  and 
expanded  adiabatically  until  the  temperature  falls  to  32°F.     (7)  [0.41]. 

7.  What  is  the  weight  of  1  cu.  ft.  of  air  when  pa  =  14.0  and  Ta  =  —  10°? 
(8)  [0.84]. 

8.  A  compressor  cylinder  is  20  in.  in  diameter  by  26-in.  stroke  double- 
acting.     Clearance  =  0.8  per   cent.,   piston  rod   =  2  in.,   r.p.m.   =  100, 
atmospheric  pressure,  pa  =  14.3,  atmospheric  temperature  =  Ta  =  60°FM 
and  gage  pressure  =  98  Ib. 

Determine  the  following : 
(a)  Compression  isothermal. 

la.  Volume  of  free  air  compressed,  cubic  feet  per  minute.     (46)  [891]. 
2a.  Volume  of  compressed  air,  cubic  feet  per  minute.     (1)  [1,144]. 
3a.  Work  of  compression,  foot-pounds  per  minute.     (1)  [3,757,000]. 
4a.  Pounds  of  cooling  water,  TI  =  50°,  T2  =  75°.     (9)  [193]. 
(6)  n  =  1.25  and  air  heated  to  100°  while  entering. 

16.  Volume  of  free  air  compressed  per  minute.     (46)  [830]. 
26.  Volume  of  cool  compressed  air  per  minute.     (1)  [106.5]. 
36,  Work  done  in  compression.     (1)  [4,658,000]. 
46.  Temperature  of  air  at  discharge.     (7)  [385°F.]. 

9.  The  cylinder  of  a  compressed-air  motor  is  18  by  24  in.,  the  r.p.m.  =  90, 
air  pressure  100  Ib.  gage.     In  the  motor  the  air  is  expanded  to  four  times  its 
original  volume  (cutoff  at  H)>  with  n  =  1.25. 

(a)  Determine  the  horsepower  and  final  temperature  when  initial  T  = 
60°F.  (3  and  7)  [hp.  =  132,  T  =  -90]. 

(6)  Determine  the  horsepower  and  final  temperature  when  initial  T  = 
212°F.  (3  and  7)  [hp.  =  132,  T  =  +17]. 

10.  Observations  on  an  air  compressor  show  the  intake  temperature  to 
be  60°F.,  the  r  =  7  and  the  discharge  temperature  =  300°F.     What  is  the 
n  during  compression? 

Hint.— Use  Eq.  (llo)  with  n  unknown.     (7)  [1.25]. 

11.  In  a  compressed-air  motor  what  percentage  of  power  will  be  gained 
by  heating  the  air  before  admission  from  60°  to  300°F.?     (2)  [46  per  cent.]. 


124 


COMPRESSED  AIR 


12.  If  air  is  delivered  into  a  motor  at  60°F.  and  the  exhaust  temperature 
is  not  to  fall  below  32°F.,  what  ratio  of  expansion  can  be  allowed?     What 
could  be  allowed  if  initial  temperature  were  300°?     n  =  1.25.     (2  and  7) 
[1.31,  8.8]. 

13.  A  compressed-air  locomotive  system  is  estimated  to  require  4,000 
cu.  ft.  per  minute  of  free  air  compressed  to  500  Ib.  gage  in  three  stages  with 
complete  cooling  between  stages. 

Assume  n  =  1.25,  pa  =  14.5,  Ta  =  60°,  vol.  eff.  =  80  per  cent.,  mech. 
eff.  =  85  per  cent,  and  r.p.m.  =  60. 

Compute  the  volume  of  piston  stroke  in  each  of  the  three  cylinders  and  the 
total  horsepower  required  of  the  steam  end.  (13  and  14)  [41.5,  12.7,  3.87, 
1,220]. 

14.  A  compressor  is  guaranteed  to  deliver  4  cu.  ft.  of  free  air  per  revolu- 
tion at  a  pressure  of  116  (absolute).     To  test  this  the  compressor  is  caused  to 
deliver  into  a  closed  system  consisting  of  a  receiver,  a  pipe  line  and  a  tank. 
Observed  conditions  are  as  follows: 


Receiver 

Pipe 

Tank 

Pressure  at  start  (ab.)  

14  5 

14  5 

14  5 

Temperatures  at  start  (F.)  

60.0 

60  0 

60  0 

Pressures  at  end   (ab.)  

116  0 

116  0 

116  0 

Temperatures  at  end   (F.) 

150  0 

90  0 

60  0 

Volumes   (cubic  feet)  .  .          .    . 

50  0 

10  0 

100  0 

How  many  revolutions  of  the  compressor  should  produce  this  effect? 
(27)  [264]. 

16.  Find  the  discharge  in  pounds  per  minute  through  a  standard  orifice 
when  d  =  2  in.,  i  =  5  in.,  t  =  600°  and  pa  =  14.0.  (21)  [8.03]. 

16.  What  diameter  of  orifice  should  be  supplied  to  test  the  delivery  of  a 
compressor  that  is  guaranteed  to  deliver  1,000  cu.  ft.  per  minute  of  free  air? 
(21)  [6.5]. 

17.  What  is  the  efficiency  of  transmission  when  air  pressure  drops  from 
100  to  90  Ib.  (gage)  in  passing  through  a  pipe  system?     (31)  [95.5]. 

18.  A  compressor  must  deliver  100  cu.  ft.  per  minute  of  compressed  air 
at  a  pressure  =  90  Ib.  gage,  at  the  terminus  of  a  pipe  3,000  ft.  long  and  3  in. 
in  diameter.     pa  =  14.4,  Ta  =  60°F. 

(a)  Assuming  a  vol.  eff.  =  75  per  cent.,  what  must  be  the  piston  displace- 
ment of  the  compressor?  [967]. 

(6)  What  pressure  is  lost  in  transmission?     (29)  [17]. 

(c)  What  horsepower  is  necessary  in  steam  end  of  compressor  if  n  =  1.25 
and  the  mech.  eff.  =  85  per  cent.?     (29  and  2)  [141]. 

(d)  What  would  be  the  efficiency  of  the  whole  system  if  air  is  applied  in 
the  motor  without  expansion,  the  efficiency  to  be  reckoned  from  steam 
engine  to  work  done  in  motor?     (6)  [27  per  cent.]. 

19.  It  is  proposed  to  convey  compressed  air  into  a  mine  a  distance  of 
5,000  ft.     The  question  arises:  Which  is  better,  a  3-in.  or  a  4-in.  pipe? 

Compare  the  propositions  financially,  using  the  following  data:  Nominal 


EXAMPLES  AND  EXERCISES  125 

capacity  of  the  plant  =  1,000  cu.  ft.  free  air  per  minute,  vol.  eff.  of  com- 
pressor =  80  per  cent.,  n  =  1.25  gage  pressure  at  compressor  =  100,  weight 
of  free  air  wa  =  0.074,  pa  =  14.36,  weight  of  3-in.  pipe  =  7.5  and  of  4-in. 
pipe  =  10.7  Ib.  per  foot.  Cost  of  pipe  in  place  =  4  cts.  per  pound.  Cost  of 
1  hp.  in  form  of  pv  log  r  for  10  hr.  per  day  for  1  year  =  $150.  Plant  runs 
24  hr.  per  day.  Rate  of  interest  =  6  per  cent.  (29)  [Economy  of  4-in. 
pipe  capitalized  =  $86,260]. 

20.  Air  enters  a  4-in.  pipe  with  60  ft.  velocity  and  80  Ib.  gage  pressure; 
the  air  pipe  is  1,500  ft.  long. 

(a)  Find  the  efficiency  of  transmission.     (31)  [91  per  cent.]. 

(6)  Find  horsepower  delivered  at  end  of  pipe  in  form  pv  log  r.     (31)  [224]. 

(c)  Find  horsepower  delivered  at  end  of  pipe  in  form  Pa  X  v.     (31)  [73.5]. 

21.  An  air  pipe  is  to  be  2,000  ft.  long  and  must  deliver  50  hp.  at  the  end 
with  a  loss  of  5  per  cent,  of  the  pv  log  r  as  measured  at  compressor.     The 
pressure  at  compressor  is  75  Ib.  gage.     pa  =  14.7.     Find  diameter  of  pipe. 
(29)  [2%]. 

22.  Modify  21  to  read:  50  hp.    .    .  with  loss  of  5  per  cent,  of  the  energy 
in  form  Pg  X  v,  where  Pg  is  gage  pressure,  and  find  diameter  of  air  pipe. 
(29)  [3H1- 

23.  In  case  21  let  pressure  at  compressor  be  250  Ib.  gage  and  find  diameter 
of  air  pipe.     (29)  [1.4]. 

24.  The  air  cylinder  of  a  compressed-air  pump  is  20  in.  in  diameter  by  30- 
in.  stroke.     The  machine  is  double-acting  and  makes  50  r.p.m.     The  cutoff 
is  to  be  so  adjusted  that  the  temperature  of  exhaust  shall  be  30°.     pa  = 
14.5  and  the  r  at  pump  =8.     n  =  1.25. 

(a)  Find  cutoff  when  initial  temperature  is  60°F.     [0.78]. 
(6)  Find  cutoff  when  initial  temperature  is  250°F.     [0.226]. 

(c)  Find  horsepower  in  case  (a).     [223]. 

(d)  Find  horsepower  in  case  (6).     [112]. 

(e)  In  case  (a)  find  efficiency  in  applying  the  pv  log  r  of  cool  air.     [55 
per  cent.]. 

(/)  In  case  (6)  find  efficiency  in  applying  the  pv  log  r  of  cool  air.  [85  per 
cent.]. 

(g)  Find  the  volumes  of  free  air  used  in  cases  (a)  and  (6).  [3,400  and 
732]. 

25.  A  compound  mine  pump  is  to  receive  air  at  150  Ib.  gage;  this  is  to  be 
reheated  from  60°  to  250°F.,  let  into  the  H.P.  cylinder  of  the  pump  and  ex- 
panded until  the  temperature  is  32°,  then  exhausted  into  an  interheater 
where  the  temperature  is  again  brought  to  250°.     It  then  goes  into  the  L.P. 
cylinder  and  is  expanded  down  to  atmospheric  pressure  =  14.5  (ab.). 

(a)  Find  point  of  cutoff  in  each  cylinder,  n  =  1.25.     [0.23  and  0.61]. 

(b)  If  the  air  is  compressed  in  two  stages  with  n  =  1.25,  what  will  be  the 
efficiency  of  the  system,  neglecting  friction  losses?     [1.06]. 

(c)  How  much  free  air  will  be  required  to  operate  the  pump  if  it  is  to 
deliver  250  hp.,  assuming  the  efficiency  of  the  pump  to  be  80  per  cent, 
reckoned  from  the  work  in  the  air  end?     [1,686]. 

(d)  If  the  pump  strokes  be  60  per  minute  and  60  in.  long,  fix  diameters  of 
air  cylinders  in  case  (c).     [23  in.  and  35  in.] 

26.  Compute  the  horsepower  of  a  motor  passing  1  Ib.  of  air  per  minute 


126  COMPRESSED  AIR 

admitted  at  200°F.  and  116  Ib.  (ab.)  r  =  8,  the  air  to  be  expanded  until 
pressure  drops  to  29  Ib.  (ab.),  r  =  2.     n  =  1.25.     (3  and  7)  [1.727]. 

27.  A  pump  to  be  operated  by  compressed  air  must  deliver  1,000  gal.  of 
water  per  minute  against  a  net  head  of  200  ft.  through  800  ft.  of  10-in.  pipe. 
The  pump  is  double-acting,  30-in.  stroke,  50  strokes  per  minute.     The  air  is 
reheated  to  275°F.  before  entering  the  pump.     The  cutoff  is  so  adjusted  that 
with  n  =  1.25  the  temperature  at  exhaust  =  36°F.     Mec.  eff.  of  pump=  80 
per  cent.     Air  pressure  at  compressor  =  80  Ib.  gage,  pa  =  14.4.    Length  of 
air  pipe  =  2,000  ft.     Permissible  loss  in  transmission  =  7  per  cent,  of  the 
pv  log  r  at  compressor.     Mec.  eff.  of  compressor  =  85  per  cent.     Vol.  eff. 
=  80  per  cent. 

(a)  Proportion  the  cylinders  of  the  pump.    [Water  14  in.,  air  26  in.]. 
(6)  Determine  the  volume  of  free  air  used.    [444]. 
(c)  Determine  the  diameter  of  air  pipe.    [3^]« 

28.  Compare  the  volume  displacement  of  two  air  compressors,  one  at 
sea  level  and  the  other  at  12,000  ft.  elevation;  the  compressors  to  handle 
the  same  weight  of  air.    {9.45  -f-  14.7]. 

29.  (a)  An  exhaust  pump  has  an  effective  displacement  of  3  cu.  ft.  per 
revolution.    How  many  revolutions  will  reduce  the  pressure  in  a  gas  tank 
from  30  to  5  Ib.  absolute,  volume  of  tank  =  400  cu.  ft.?     (15)  [239]. 

(6)  If  the  pump  is  delivering  the  gas  under  a  constant  pressure  of  30  Ib. 
(ab.)  what  is  the  maximum  rate  of  work  done  by  the  pump — foot-pounds 
per  revolution?  n  =  1.25.  (15)  [5,433]. 

30.  An  air-lift  pump  is  to  be  designed  to  elevate  gravel  from  a  submerged 
bed.     Specifications  as  follows : 

Depth  of  submergence  =  50  ft. ;  lift  above  water  surface  =  10  ft. ;  volume 
lifted  to  be  Y±  gravel  and  %  sea  water;  specific  gravity  of  gravel  =  3;  weight 
of  sea  water  =  65  Ib.  per  cubic  foot;  volume  of  gravel  =  1  cu.  yd.  per  minute. 

(a)  Determine  the  ratio  -Qt  Q  =  volume  of  mixed  water  and  gravel. 

(6)  Determine  the  ratio  of  compression  and  horsepower  of  compressor, 
(c)  Recommend  diameters  for  water  pipe  and  for  air  pipe.     (41). 


TABLES 


NOTES  ON  TABLE  i 

The  table  is  designed  to  reduce  the  labor  of  solution  of  formulas  12,  12 a,  Sd 
and  10. 

When  the  weight  of  air  passed  and  its  initial  temperature  are  known,  the 
table  covers  all  conditions  such  as  elevation  above  sea  level,  reheating  and  com- 
pounding, but  it  does  not  include  the  effect  of  friction  and  clearance. 

In  compound  compression  the  same  weight  goes  through  each  cylinder.  Then 
knowing  the  initial  i  and  the  r  for  each  cylinder,  find  from  the  table  the  work  done 
in  each  cylinder  and  add.  Usually  the  r  and  t  are  assumed  the  same  in  each 
cylinder.  In  that  case  take  out  the  work  for  one  stage  and  multiply  by  the 
number  of  stages. 

The  columns  headed  "Work  Factor"  are  applicable  in  cases  of  expansion, 
only  when  the  expansion  is  complete,  that  is,  when  final  pressure  in  the  cylinder 
is  equal  that  outside.  (In  free  air  or  in  a  receiver.)  • 

Example. — Air  is  received  at  such  a  pressure  that  r  =  8.  What  should  be.  the 
cutoff  in  order  that  the  temperature  drop  from  60°  to  32°F.  when  expansion  is 
adiabatic? 

The  ratio  of  absolute  temperatures  is  1.057  which  by  linea  interpolation  corre- 
sponds to  a  volume  ratio  0.871  or  cutoff  is  at  J^. 

What  would  be  the  pressure  at  exhaust? 

The  two  ratios  above  are  in  the  horizontal  line  with  -  =  .825  therefore  the 

final  pressure  =  .825  X  initial  pressure. 

To  find  the  foot-pounds  per  pound  of  air,  multiply  the  number  opposite  r  in 
columns  7,  8  or  n  as  the  case  may  be  by  the  absolute  lower  temperature. 

To  find  the  weight  compressed,  go  into  Table  II  with  known  atmospheric  con- 
ditions and  cubic  feet  capacity  of  the  machine. 

To  find  the  horse-power  per  100  of  air  per  minute  multiply  the  number  oppo- 
site r  in  columns  9, 10  or  12,  as  the  case  may  be,  by  the  absolute  lower  temperature. 


128 


TABLE  I. — GENERAL  TABLE  RELATING  TO  AIR  COMPRESSION 
AND  EXPANSION 


o 

+L 

Ratio  of 
Less  to 
Greater 

Ratio  of 
Greater  to 
Less  Tem- 

Work Factor. 
Air  Heated  by  Compression 

Work  Factor 
for  Isothermal 
Compression 

Volume  — 

perature  — 

§ 

Pg 

_ 

Tempera- 

Tempera- 

K =  51.17  —  — 

H.P.  Fac- 

O Q 

f  Compi 
Expansi 

I? 

31 

tures 
Changing 

I 

tures  Ab- 
solute 

n-i 

1  n  -  i 

tor  per  100 
Pounds  per 
Minute 

fh 

il 

O  4) 

X  I    n  -  i        j 

o 
1 

*o  3 
o'o 

'£  ^ 

?-(*)" 

HO  ' 

Factor  K  for  one 
pound 

K 
=  330 

aff 

y 

r/ 

J 

II   ^H 

ii 

& 

n  = 

n  = 

^ 

K 

1.25 

1.  14 

330 

r 

I 

Vt 

3 

fc 

fa 

Ft.-Lbs. 

Ft.-Lbs. 

H.P. 

H.P. 

Ft.-Lbs. 

H.P. 

r 

Vl 

n 

ti 

* 

i 

2 

3 

4 

5 

6 

7 

8 

9 

10 

ii 

12 

I  .  OOOO 

I.OOO 

I.OOO 

"l.OOO 

I.OOO 

o.o 

0.0 

.O 

.0 

0.0 

.O 

.  i 

.909I 

.927 

.935 

1.019 

1.028 

5.131 

5.140 

-0155 

-0155 

5.068 

•0153 

.  2 

.8333 

.862 

.877 

1.037 

1.054 

9.863 

9.932 

.0298 

.0301 

9.694 

.0293 

.3 

.7692 

.812 

.830 

1.054 

1.079 

14.329 

14.450 

•0434 

•0437 

I3-950 

.0422 

•  4 

•7143 

.764 

.787 

1.070 

1.103 

18.503 

18.766 

.0560 

.0568 

17.890 

.0542 

•  5 

.6667 

•  723 

•  750 

1.085 

1.1*5 

22.465 

22.827 

.0680 

.0691 

21-559 

.0653 

.6 

.6250 

.687 

.717 

I.IOO 

1.146 

26.186 

26.704 

•0793 

.0809 

24.991 

•0757 

•  7 

.5882 

.654 

.686 

1.  112 

1.166 

29.775 

30.4I7 

.0902 

.0921 

28.214 

•0855 

.8 

•5555 

.625 

•659 

I.I25 

1.186 

33.178 

33.985 

•  1005 

.1029 

3L252 

.0947 

-9 

•5263 

.598 

•634 

I-I37 

1.205 

36.421 

37.422 

.1104 

.1134 

34.127 

.1034 

.0 

.5000 

•574 

.612 

I.  149 

1.223 

39.530 

40.733 

.1198 

•1235 

36-855 

.1117 

2.1 

.4762 

•552 

•590 

1.160 

1.240 

42.536 

43.897 

.1289 

.1330 

39-450 

.1196 

2.2 

•4545 

•532 

•571 

1.171 

1-259 

45.407 

46.988 

.1376 

.1424 

41.912 

.1270 

2.1 

•  4348 

.514 

-553 

1.181 

1.273 

48.199 

49-97° 

.1461 

.1514 

44.287 

.1342 

3.4 

.4166 

.496 

•537 

1.191 

1.289 

50.884 

52.878 

.1542 

.1602 

46.548 

.1411 

2-5 

.4000 

.480 

-522 

i.  202 

1.304 

53.462 

55.676 

.  1620 

.1687 

48.720 

.1476 

2.6 

.3846 

.466 

.508 

I.  211 

1.319 

55.988 

58.402 

.1697 

.1769 

50.805 

.1539 

2.7 

•37°4 

.452 

•493 

1.220 

1-334 

58.434 

61.054 

.1771 

.1850 

52.811 

.  I6OO 

2.8 

•3571 

•439 

.481 

1.229 

1.348 

60.800 

63-651 

.1843 

.1929 

54-745 

.1659 

2.9 

.3448 

.427 

.469 

1.237 

1.362 

63  .  086 

66.175 

.1912 

.2006 

56.612 

•  J7J5 

3.0 

•3333 

•  415 

•  458 

1.246 

1-375 

65-319 

68.626 

.1979 

.2080 

58.414 

.1770 

3.1 

•  3226 

-405 

-448 

1.254 

1.388 

67  .  499 

71.158 

.2045 

.2156 

60.157 

.1823 

3-2 

•3125 

•394 

.438 

1.262 

1.401 

69.626 

73.400 

.2110 

.2224 

61.845 

.1874 

3/3 

•  303° 

•385 

.428 

1.270 

1.414 

71.  700 

75.686 

.2173 

.2294 

63.481 

.1924 

3-4 

.  2941 

•376 

•  419 

1.277 

1.426 

73-720 

77.936 

.2234 

.2362 

65.087 

.1972 

3-5 

•2857 

•367 

.411 

1.285 

1-438 

75-688 

80.131 

.2294 

.2428 

66.610 

.2019 

3-6 

•2778 

•359 

-403 

1.292 

1.450 

77.628 

82.307 

.2352 

.2494 

68.108 

.2064 

3-7 

.2703 

•351 

•395 

1.299 

1.461 

79.516 

84.411 

.2410 

•2557 

69.564 

.2108 

3-8 

•  2632 

•343 

•  388 

1.306 

1-473 

81.350 

86.496 

.2465 

.2621 

70.982 

.2151 

3-9 

.2564 

•337 

-381 

I-3I3 

1.484 

83-158 

88.544 

.2520 

.2683 

72.364 

.2193 

4.0 
4.1 

.2500 
.2439 

•33° 
•323 

•374 
•367 

1-319 
1.326 

1.506 

84.939 
86.694 

90.510 
92.472 

•2574 
.2627 

•2743 
.2802 

73-7io 
75-023 

•2234 
•  2274 

4-2 

-2381 

-361 

1-332 

1.516 

88-395 

94-434 

.2678 

.2862 

76.304 

.2312 

4-3 

.2326 

•311 

•355 

1-339 

1.526 

90.043 

96.346 

.2729 

.2919 

77-555 

•235° 

4-4 

.2273 

.306 

•349 

1-345 

1-537 

91.691 

98.202 

•2779 

.2976 

78.776 

.2387 

4-5 

.2222 

.300 

•344 

I-35I 

i-547 

93-312 

100.012 

.2828 

•3031 

79.972 

.2424 

4.6 

.2174 

•295 

.338 

i-357 

1-557 

94.882 

101.823 

•2875 

•3085 

81.141 

•2459 

4-7 

.2128 

290 

-333 

1-363 

1-566 

96.424 

103.616 

.2922 

.3140 

82.284 

.2494 

4.8 

.2083 

•285 

.328 

1.368 

1.576 

97.966 

105.371 

.2969 

•3193 

83.404 

.2528 

129 


130 


COMPRESSED  AIR 


TABLE  1. —(Continued} 


I 

2 

3 

4 

5 

6 

7 

8 

9 

10 

II 

12 

4.9 

.2041 

.280 

•  324 

1-374 

1-58 

99.481 

107.109 

.301 

•  3246 

84.500 

.2561 

5-o 

.2000 

.276 

•3i9 

1.380 

1-59 

100.943 

108.811 

•3059 

•3297 

85.574 

•2593 

5-i 

.1961 

.272 

•3i5 

1-385 

i.  60 

102.405 

110.493 

-3103 

•3348 

86.627 

.2625 

5-2 

.1923 

.267 

.310 

1-391 

1.613 

103.841 

112.157 

-3I47 

•3398 

87.660 

•2657 

5-3 

.l88" 

.263 

.306 

1.396 

1.622 

105.260 

113.830 

.3180 

•3449 

88.673 

.2687 

5-4 

.1852 

•259 

.302 

1.401 

1.63 

106.673 

115.440 

•3232 

.3498 

89.666 

.2717 

5-5 

.l8l8 

.256 

.298 

1.406 

1.640 

108.013 

117.010 

•3273 

.3546 

90.642 

.2747 

5-6 

.I786 

.252 

.294 

1.411 

1.648 

109-353 

118.570 

.3314 

•3593 

91.600 

2776 

5-7 

•1754 

.248 

.291 

1.416 

1-657 

110.683 

I2O.  114 

•3354 

.3640 

92.541 

2805 

5-8 

.1722 

•245 

.287 

1.421 

1.665 

112.003 

121.632 

•3394 

.3686 

93.466 

2833 

5-9 

.1695 

.242 

.284 

1.426 

1.673 

113-305 

123.150 

•3433 

•3732 

94-375 

2860 

6.0 

.1667 

.238 

.280 

i  -431 

1.681 

114.581 

124.640 

•3472 

•3777 

95-271 

2887 

6.1 

.1639 

•235 

.277 

1.436 

1.689 

115.831 

I26.II3 

•3510 

3822 

96.147 

2914 

6.2 

.1613 

.232 

.274 

1.440 

1.697 

117.080 

127.576 

•3548 

3866 

97.012 

2940 

6-3 

.1587 

.229 

.271 

1-445 

1-705 

118.303 

129.030 

•3585 

3910 

97-863 

2966 

6.4 

.1562 

.226 

.268 

1.449 

i-7i3 

"9.573 

130.466 

.3622 

3953 

98.700 

2991 

6.5 

•I538 

.223 

.265 

1-454 

1.721 

120.723 

131.880 

•3658 

3997 

99.524 

3016 

6.6 

•1515 

.221 

.262 

1.458 

1.728 

121.920 

I33-300 

•3694 

4039 

100.336 

3040 

6.7 

.1492 

.219 

•259 

1.464 

1.736 

123.063 

134.710 

•3729 

.4082 

101.134 

3065 

6.8 

.1471 

.216 

.256 

1.467 

1-744 

124.205 

136.090 

•3764 

.4124 

101.920 

3088 

6.9 

.1449 

.213 

•  254 

1.471 

i.75i 

125.348 

I37.450 

•3799 

4165 

IO2  .  700 

3112 

7.0 

.1428 

.211 

-251 

1.476 

I.758 

126.492 

138.800 

.3833 

4206 

103.465 

3135 

7-i 

.1408 

.208 

.249 

1.480 

1.766 

127.608 

I4O.  120 

.3867 

4246 

164.219 

3158 

7.2 

.1389 

.206 

.246 

1.484 

1-773 

128.708 

141.430 

.3900 

4286 

104.963 

3l8l 

7-3 

.1370 

.204 

.244 

1.488 

1.780 

129.789 

142.710 

-3933 

4327 

105.696 

3203 

7-4 

•I351 

.202 

.241 

1.492 

1.787 

130.878 

143-979 

.3966 

4363 

106.420 

3225 

7-5 

•1333 

.199 

•239 

1.496 

1.794 

131.941 

145.239 

•3998 

44oi 

107.133 

3246 

7.6 

•  1316 

.197 

•237 

1.500 

1.801 

132.995 

146.489 

-4030 

4439 

107.837 

3268 

7-7 

.1299 

•195 

•235 

1.504 

1.807 

134-043 

147.732 

.4062 

4477 

108.539 

3289 

7.8 

.1282 

•193 

•233 

1.508 

1.814 

135-063 

148.976 

•4093 

45H 

109.219 

3310 

7-9 

.1266 

.191 

.231 

1.512 

1.821 

136.091 

150.217 

.4124 

4552 

109.896 

333° 

8.0 

.1250 

.l89 

.228 

1-516 

1.828 

137.110 

I5L427 

•4155 

4589 

110.565 

335° 

8.1 

.1236 

.188 

.226 

I.5I9 

1.834 

138.111 

152.633 

.4185 

4625 

III.225 

337° 

8.2 

.  I22O 

.186 

.224 

!-523 

1.841 

139-093 

I53.823 

.4215 

4661 

III.875 

339° 

8.3 

.1205 

.184 

.223 

1.527 

1.847 

140.076 

155.010 

•4245 

4698 

112.522 

34io 

8.4 

.1190 

.182 

.221 

1.531 

1.854 

141.060 

156.178 

•4275 

4733 

II3.I58 

3429 

8-5 

.1176 

.l8o 

.219 

1-534 

1.861 

142.017 

157.348 

•4304 

4768 

H3.788 

3448 

8.6 

.1163 

.179 

.217 

1.538 

1.867 

142.974 

158.508 

•4333 

4804 

II4.4IO 

3465 

8.7 

.  1  149 

.177 

.215 

i.54i 

1.873 

143-931 

159.658 

.4362 

4838 

115.023 

3487 

8.8 

.1136 

.176 

.214 

J-545 

1.879 

144.862 

160.800 

•439° 

4873 

I5-633 

35°4 

8.9 

.1124 

.174 

.212 

1.548 

1.885 

45.78o 

161.927 

4418 

4906 

16.233 

3522 

9.0 

.1111 

.172 

.210 

L552 

1.891 

46.700 

163.041 

4446 

4941 

16.827 

3540 

9.1 

.1099 

.171 

.208 

1-555 

1.897 

47.627 

164.147 

4474 

4974 

I7.4I5 

355s 

9.2 

.1087 

.170 

.207 

i  -559 

1.903 

48-557 

165.236 

45°2 

5007 

I7.996 

3576 

9-3 

.1072 

.168 

•205 

.562 

1.909 

49-554 

166.334 

4532 

5°4i 

18.571 

3593 

9.4 

.  1064 

.167 

.204 

.565 

•915 

50.312 

167.431 

4555 

5°74 

19.138 

3610 

9-5 

.  1058 

.165 

.202 

•569 

.921 

51.188 

168.520 

4582 

5107 

19.702 

3627 

9.6 

.1042 

.164 

.201 

1.5721.927 

52.066 

169.589 

4609 

5139 

20.259 

3644 

9-7 

:io3i 

.162 

.199 

i-575I-933 

52.944 

170.650 

4635 

5i7i 

2O.8lO 

3661 

9.8 

.  1020 

.l6l 

.198 

1.5781-939 

53-794 

171.700 

4661 

5213 

21-355 

3677 

9.9 

.  IOIO 

,l6o 

.I96 

1.5821.944 

54.645 

172.754 

4686 

5235 

21.895 

3693 

10.  0 

.IOOO 

•159 

•195 

1.5851.950 

55-495 

I73.789 

4712 

5266 

22.429 

3710 

TABLES 


131 


NOTES  ON  TABLE  II 

The  purpose  of  this  table  is  to  determine  the  weight  of  air  compressed  by  a 
machine  of  known  cubic  feet  capacity.  It  is  to  be  used  in  connection  with  Table 
I  for  determining  power  or  work. 

The  barometric  readings  and  elevations  are  made  out  for  a  uniform  tempera- 
ture of  60°  F.  and  are  subject  to  slight  errors  but  not  enough  to  materially  affect 
results.  Table  V  gives  more  accurately  the  relation  between  elevation  tem- 
perature and  pressure. 

TABLE  II.— WEIGHTS  or  FREE  AIR  UNDER  VARIOUS  CONDITIONS 


it 
Pi 

ric  Pressure 

Weight  of  One  Cubic  Foot  at  Given  ' 
Temperature  (Fahr.) 

Rj 
£  0 

'§  1^ 

"ft 

0  C 

ft  C 

8 

-20° 

00° 

20° 

40° 

60° 

80° 

100° 

I  ! 

< 

g 

•*-> 

I 

2 

3 

4 

5 

6 

7 

8     9 

10 

30.52 

15.0 

.09211 

.08811 

.08444 

.08108 

.07796 

.07508 

.07240 

-600 

30.32 

14.9 

.09150 

.08753 

.08388 

.08054 

•07744 

.07458 

.07192 

—400 

30.12 

14-8 

.  09089 

.08694 

•08331 

.  08000 

•07693 

.  07408 

.07144 

—  2OO 

29.91 

14.7 

.09027 

.08635 

.08275 

•07945 

.07640 

•07358 

.07095 

00 

29.71 

14.6 

.08965 

.08576 

.08219 

.07895 

.07589 

.07308 

•07047 

200 

29.50 

14-5 

.08903 

.08517 

.08163 

•07837 

•07536 

.07258 

.06999 

400 

29.30 

14.4 

.08842 

.08458 

.08107 

•07783 

.07484 

.07208 

.06950 

600 

29.10 

14-3 

.08781 

.  08400 

.08050 

.07729 

.07432 

.07158 

.06902 

800 

28.90 

14.2 

.08719 

.08341 

.07994 

•07675 

•07380 

.07108 

.06854 

1000 

28.69 

I4.I 

.08659 

.08282 

.07938 

.07621 

•07329 

•07058 

.06806 

I20O 

28.49 

14-0 

•08597 

.08224 

.07882 

•07567 

.07277 

.07008 

•06758 

1400 

28.28 

13-9 

•08535 

.08165 

.07825 

.07513 

.07225 

.06957 

.06709 

l6oO 

28.08 

13-8 

.08474 

.08106 

.07769 

•07459 

•07173 

.06907 

.06661 

1800 

27.88 

13-7 

.08412 

.  08048 

•07713 

.07405 

.07120 

.06857 

.06612 

200O 

27.67 

13-6 

.08351 

.07989 

.07656 

•0735° 

.07068 

.  06807 

.06564 

2IOO 

27.47 

13-5 

.08289 

.07930 

.07600 

.07296 

.07016 

•06757 

.06516 

2300 

27.27 

13-4 

.08228 

.07871 

•07544 

.07242 

.06965 

.06707 

.06468 

250O 

27.06 

13-3 

.08167 

.07813 

.07487 

.07189 

•06913 

•06657 

.06420 

270O 

26.86 

13.2 

.08106 

.07754 

•07431 

•07135 

.06861 

.06607 

.06371 

29OO 

26.66 

I3-I 

.  08044 

.07695 

•07375 

.07080 

.06809 

•o6557 

.06323 

3100 

26.45 

13.0 

.07983 

.07637 

.07319 

.07026 

.06757 

.06507 

.06274 

33°0 

26.25 

12.9 

.07921 

.07578 

.07262 

.06972 

.06705 

.06457 

.06226 

35°° 

26.05 

12.8 

.07860 

•07518 

.07206 

.06918 

.06652 

.06407 

.06178 

3700 

25.84 

12.7 

.07798 

.07460 

.07150 

.06862 

.06600 

.o6357 

.06130 

4000 

25.64 

12.6 

.07737 

.07401 

.07094 

.06810 

.06549 

.06307 

.06082 

4200 

25.44 

12.5 

.07676 

•07343 

.07038 

.06756 

.06497 

.06257 

.06033 

4400 

25.23 

12.4 

.07615 

.07284 

.06981 

.06702 

.  06445 

.06207 

•05985 

4600 

132 


COMPRESSED  AIR 


TABLE  II.— (Continued) 


I 

2     3 

4 

5 

6 

7 

8 

9 

10 

25-03 
24.83 

24.62 

12.3 

12.2 
12.  I 

•07553 
.07492 
.07430 

.07225 
.07166 
.07108 

.06925 
.06868 
.06812 

.  06648 
•06594 
.06540 

.06393 
•06341 
.06289 

.06157 
.06107 
•06057 

•05937 
.05889 
.  05840 

4800 
5000 
5200 

24.42 

24.22 
24.01 

12.0 
II-9 

ii.  8 

.07369 
.07307 
.07246 

.07049 
.06990 
.06932 

.06756 
.  06699 
•  06643 

.  06486 
.06432 
•06378 

.06237 
.06185 
.06133 

.06007 
•05957 
•05907 

.05792 

•05744 
.05696 

54oo 
5600 
5800 

23.81 

23.60 

23.40 

"'£ 
ii.  6 

n-5 

.07184 
.07123 
.07061 

.06873 
.06812 
•06755 

.06587 
-06530 
.06474 

.06324 
.06270 
.06216 

.06081 
.06029 
•05977 

•05857 
.05807 

•05757 

.05647 
•05599 
-05551 

6100 
6300 
6500 

23.20 

22.99 
22.79 

11.4 
n-3 

II.  2 

.07000 
.06938 
.06877 

.06693 
.06638 
-06579 

.06418 
.  06362 
•  06305 

.06161 
.06108 
.06054 

•05925 
•05873 
.05821 

.05707 
.05656 
.05606 

•05502 
•05454 
.05406 

6800 
7100 
7300 

22.59 
22.38 
22.18 

II.  I 
II.  0 
10.9 

.06816 
.06754 
.06692 

.06520 
.  06462 
.06403 

.06249 
.06193 
.06136 

.06000 

•05945 
.05891 

•05769 
•05717 
.05665 

•05556 
.05506 
•05456 

.05358 
•0531° 
.05261 

7600 
7900 
8100 

21.98 
21-77 
21-57 

10.8 
10.7 
10.6 

.06632 
.06571 
.06510 

•  06344 
.06285 
.06226 

.06080 
.06024 
.05968 

-05837 
•05783 
.05729 

•05613 
•05561 

•05509 

.05406 
•05356 
.05306 

•05213 
•05164 
.05116 

8400 
8600 
8900 

21.37 

21  .  ID 
20.96 

10.5 
10.4 
10.3 

.  06448 
.06386 
.06325 

.06168 
.06109 
.06050 

.05911 
•05855 
•05799 

•05675 
.05621 

•05567 

•05457 
•05405 
•05353 

•05256 
.05206 
•05156 

.05068 
.05020 
.04972 

9100 
9400 
9600 

20.76 

20.55 
20.35 

10.2 
10.  I 
10.  O 

.06263 
.06202 
.06141 

.05991 

•05933 
•05874 

-05743 
.05686 
.05630 

•05513 
•05459 
•05405 

•05301 
.05249 
•05198 

.05106 
.05056 
.05006 

.04923 
.04875 
.04827 

9900 

IOIOO 

10400 

20.  15 
19.94 
19.74 

9-9 
9.8 

9-7 

.06079 
.06017 
•05956 

.05816 

•05757 
.05698 

•05572 
•05517 
.05461 

•05351 
.05297 

•05243 

.05146 

•05094 
.05041 

.04956 
.  04906 
.04856 

.04779 
•04730 
.  04682 

10700 

IIOOO 
II2OO 

19-53 
19-33 
19-13 

9.6 
9-5 
9-4 

.05894 

•05833 
.05772 

•05639 
.05580 
-05522 

.05404 
•05348 
.05292 

.05188 

-05134 
.05081 

.  04990 

•04937 
.04886 

.  04806 
.04756 
.04706 

•04633 

•04585 
•04538 

II500 
IISOO 
1  2  100 

18.93 
18.72 
18.52 

9-3 
9.2 
9.1 

.05711 
.05649 
•o5587 

•05463 

.05404 

•05345 

.06236 

•05179 
•05123 

.05027 
.04972 
.04918 

.  04834 
.04782 

.0473° 

•04655 
.  04605 

•04555 

.  04489 
.  04440 
.04392 

12400 
12700 
13000 

18.31 

9.0 

•05526 

.05286 

.05067 

.  04864 

.04678 

•04505 

•  04344 

13400 

NOTE  ON  TABLE  III 
The  table  is  designed  to  compute  readily  weights  of  compressed  air  by  formula 
•     If  p  is  given  in  pounds  per  square  inch  the  formula 


becomes  .  = 


. 
53-17  Xt 


TABLE  III. — WEIGHTS  OF  COMPRESSED  AIR 

Pounds  per  Cubic  Foot 

The  Ratio  ^  is  for  absolute  pressure  in  pounds  per  square  inch  and  absolute 
temperature  Fahrenheit.     (See  Note  at  foot  of  previous  page.) 


P 


P 

t 

w 

P 
t 

•w 

i 

t 

w 

I 

t 

w 

.000 

0  .  OOOO 

•255 

.6906 

.510 

•3813 

.765 

2.0718 

.005 

•0135 

.260 

.7041 

•515 

•3947 

.770 

2.0853 

.010 

.0271 

.265 

.7177 

.520 

.4083 

•775 

2  .  0988 

.015 

.0406 

.270 

.7312 

•525 

.4219 

.780 

2.  1125 

.020 

.0542 

.275 

•7447 

-530 

•4355 

.785 

2.1260 

.025 

.0677 

.280 

•7583 

•535 

.4490 

.790 

2-1395 

.030 

.0813 

.285 

.7719 

-540 

.4625 

•795 

2.153° 

•035 

.0948 

.290 

.7852 

•545 

.4760 

.800 

2.1665 

.040 

.1083 

•295 

-  .7989 

•550 

•  4895 

•805 

2.1798 

•  045 

.1218 

.300 

.8125 

•555 

•5030 

.8lO 

2.1950 

.050 

.1354 

•3°5 

.8260 

-560 

.5166 

.815 

2.2071 

•055 

.1489 

.310 

•8395 

•565 

•5312 

.820 

2.2207 

.000 

.1625 

•3i5 

•8531 

•570 

•5437 

.825 

2.2343 

.065 

.1760 

.320 

.8666 

•575 

•5572 

.830 

2  .  2480 

.070 

.1896 

•325 

.8801 

.580 

-5707 

.835 

2.2615 

•075 

.2031 

•33° 

•8937 

•585 

•5843 

.840 

2.2750 

.080 

.2166 

•335 

.9072 

•590 

.5980 

.845 

2  .  2885 

.085 

.2302 

•  340 

.9208 

•595 

.6115 

•850 

2.3020 

.000 

•2437 

•345 

•9343 

.600 

.6250 

.855 

2-3I55 

•095 

•2573 

•35° 

.9478 

.605 

-6385 

.860 

2.3290 

.100 

.2708 

•355 

.9613 

.610 

.6520 

.865 

2.3425 

.105 

•2843 

.360 

•9749 

.615 

.6654 

.870 

2.356I 

.110 

.2979 

•365 

.9884 

.620 

.6792 

•875 

2.3698, 

•"5 

•3IJ4 

•37° 

I.OO2O 

.625 

.6927 

.880 

2.3833 

.  120 

-3250 

•375 

I-OI55 

.630 

.7062 

.885 

2.3970 

•125 

.3385 

.380 

I  .0290 

•635 

.7198 

.890 

2.4105 

.130 

.3520 

•385 

1.0425 

.640 

•7333 

.895 

2.4240 

•135 

•3656 

•390 

1.0561 

-645 

.7468 

.900 

2-4375 

.  I4O 

.3792 

•395 

1.0697 

.650 

.7603 

•90S 

2.4510 

•145 

.3927 

.400 

1-0833 

•655 

•7739 

.910 

2  •  4645 

.150 

.4062 

•405 

1.0968 

.660 

.7875 

•915 

2.4780 

•155 

.4197 

.410 

I.II03 

.665 

.8010 

.920 

2.4917 

.160 

-4333 

•415 

I.  1240 

.670 

•  8145 

•925 

2.5052 

•  165 

.4468 

.420 

I-I375 

.675 

.8280 

•93° 

2.5187 

.170 

.4603 

•425 

I.I5IO 

.680 

•8415 

•935 

2-5323 

•175 

•4739 

•  43° 

1.1645 

.685 

•8550 

.940 

2-5459 

.180 

•4875 

•435 

1.1780 

.690 

.8680 

•945 

2  5594 

.185 

.5010 

.440 

I.  1917 

•695 

.8822 

•950 

2-5730 

.190 

•SMS 

•445 

I.  2052 

.700 

.8959 

•955 

2-5865 

•195 

.5281 

•45° 

I.2I77 

•705 

.9094 

.960 

2.6000 

.200 

.5416 

•455 

1.2323 

.710 

.9229 

•965 

2  6135 

.205 

•5551 

.460 

1-2457 

•715 

•9365 

.970 

2.6270 

.210 

.5687 

•465 

1-2594 

.720 

.9500 

•975 

2  .  6405 

•215 

.5822 

.470 

1.2730 

•725 

•9635 

.980 

2.6541 

.220 

•5958 

•475 

I  .  2865 

.730 

.9770 

•985 

2  6670 

.225 

.6094 

.480 

I  .  3000 

•735 

1.9905 

990 

2.6813 

.230 

.6229 

.485 

I-3I35 

.740 

2  .  0042 

•995 

2  .  6949 

•235 

.6364 

.490 

1.3270 

•745 

2.0177 

i  .000 

2  7084 

.240, 

.6499 

•495 

I.34I6 

•750 

2.0312 

•245 

•6635 

.500 

1-3542 

•755 

2  .  0448 

.250 

.6771 

•50S 

1.3677 

.760 

2.0^82 

133 


TABLE  Ilia— GIVING  THE  VALUES  OF  "K"  AND  "H"  CORRESPONDING  TO  EACH 


Temperature, 
Degrees 
Fahrenheit 

Values  of  K 

Values  of  H 

Temperature, 
Degrees 
Fahrenheit 

Values  of  K 

Values  of  H 

Temperature, 
Degrees 
Fahrenheit 

Values  of  K 

Values  of  H 

Temperature, 
Degrees 
Fahrenheit 

Values  of  K 

Values  of  H 

Temperature, 
Degrees 
Fahrenheit 

Values  of  K 

Values  of  H 

-30 

.6082 

.0099 

17 

.6132 

.0941 

64 

.6188 

.5962 

in 

.6251 

2.654 

158 

-6323 

9.177 

—  29 

.6083 

.0105 

18 

.6133 

.0983 

65 

.6189 

.6175 

112 

.6253 

2.731 

159 

-6325 

9.400 

-28 

.6084 

.0111 

19 

.6134 

.1028 

66 

.6190 

.6393 

H3 

6255 

2.811 

160 

.6326 

9.628 

-27 

.6085 

.0117 

20 

-6135 

•  1074 

67 

.6192 

.6617 

114 

6256 

2.892 

161 

.6328 

9.860 

-26 

.6086 

.0123 

21 

.6136 

.1122 

68 

•  6193 

.6848 

US 

6257 

2.976 

162 

.6330 

IO.IO 

-25 

.6087 

.0130 

22 

.6137 

.1172 

69 

.6194 

.7086 

116 

.6258 

3.061 

163 

-6331 

10.34 

-24 

.6088 

.0137 

23 

.6139 

.1224 

70 

.6196 

•  7332 

117 

6260 

3.149 

164 

.6333 

10.59 

-23 

.6089 

.0144 

24 

.6140 

.1279 

71 

.6197 

.7585 

118 

.6261 

3.239 

165 

.6335 

10.84 

—  22 

.6090 

.0152 

25 

.6141 

.1336 

72 

.6198 

.7846 

119 

.6263 

3-331 

166 

.6336 

ii  .10 

—  21 

.6091 

.0160 

26 

.6142 

.1396 

73 

.6199 

.8114 

120 

.6264 

3.425 

167 

.6338 

11.36 

—  20 

.6092 

.0168 

27 

.6143 

.1458 

74 

.6201 

.8391 

121 

6266 

3.522 

168 

-6340 

ii  .63 

-19 

.6093 

.0177 

28 

.6144 

.1523 

75 

.6202 

.8676 

122 

.6267 

3.621 

169 

-6341 

ii  .90 

-18 

.6094 

.0186 

29 

.6146 

.1590 

76 

.6203 

.8969 

123 

6269 

3.722 

170 

.6343 

12  .18 

-17 

.6095 

.0196 

30 

.6147 

.1661 

77 

.6205 

.9271 

124 

.6270 

3.826 

171 

•  6345 

12.46 

-16 

.6096 

.0206 

31 

.6148 

•  1734 

78 

.6206 

.9585 

125 

.6272 

3-933 

172 

.6346 

12.75 

-IS 

.6097 

.0216 

32 

.6149 

.1811 

79 

.6207 

.9906 

126 

.6273 

4.042 

173 

-6349 

13.04 

-14 

.6098 

.0227 

33 

.6150 

.1884 

80 

.6209 

1.024 

127 

.6275 

4-153 

174 

-6350 

13-34 

-13 

.6099 

.0238 

34 

.6151 

.1960 

81 

.6210 

1.057 

128 

.6276 

4.267 

175 

-6352 

13.65 

—  12 

.6100 

.0250 

35 

.6153 

.2039 

82 

.6211 

1.092 

129 

.6278 

4-384 

176 

.6353 

13.96 

—  II 

.6101 

.0262 

36 

.6154 

.2120 

83 

.6213 

I.I28 

130 

.6279 

4-503 

177 

.6355 

14.28 

—  10 

.6102 

.0275 

37 

.6155 

.2205 

84 

.6214 

1  .165 

131 

.6281 

4-625 

178 

-6357 

14.60 

-  9 

.6103 

.0289 

38 

.6156 

.2292 

85 

.6215 

1.203 

132 

.6282 

4-750 

179 

.6359 

14-92 

-  8 

.6104 

.0303 

39 

.615? 

.2382 

86 

.6217 

1.242 

133 

.6284 

4-877 

180 

.6360 

15.27 

-  7 

.6105 

.0317 

40 

.6158 

.2476 

87 

.6218 

1.282 

134 

.6285 

5.008 

181 

.6362 

15.62 

-  6 

.6107 

.0332 

41 

.6160 

.2572 

88 

.6219 

1.324 

135 

.6287 

5.142 

182 

-6364 

15-97 

-  5 

.6108 

.0348 

42 

.6161 

.2673 

89 

.6221 

1.366 

136 

.6288 

5.280 

183 

-6365 

16.32 

-  4 

.6109 

.0365 

43 

.6162 

.2776 

90 

.6222 

1.410 

137 

.6290 

5.420 

184 

.6367 

16.68 

-  3 

.6110 

.0382 

44 

.6163 

.2883 

91 

.6223 

1-455 

138 

.6291 

5.563 

185 

.6369 

17.05 

—  2 

.6111 

.0400 

45 

.6164 

.2994 

92 

.6225 

I.50I 

139 

.6293 

5-709 

1  86 

•  6371 

17-43 

—  I 

6112 

.0419 

46 

.6166 

.3109 

93 

.6226 

1.548 

140 

.6294 

5-859 

187 

.6373 

17.81 

0 

.6113 

.0439 

47 

.6167 

.3227 

94 

.6227 

t.597 

141 

.6296 

6.  on 

188 

•  6374 

18.20 

+  I 

.6114 

.0459 

48 

.6168 

.3350 

95 

.6229 

1.647 

142 

.6298 

6.167 

189 

.6376 

18.59 

2 

.6115 

.0481 

49 

.6169 

•  3477 

96 

.6230 

1.698 

143 

.6299 

6.327 

190 

-6377 

19.00 

3 

.6116 

.0503 

50 

.6170 

.3608 

97 

.6232 

I.75I 

144 

.6301 

6.490 

191 

.6380 

19.41 

4 

.6117 

.0526 

51 

.6172 

.3743 

98 

.6233 

i  .805 

145 

.6302 

6.656 

192 

.6381 

19.83 

5 

.6118 

.0551 

52 

.6173 

.3883 

99 

.6234 

1.861 

146 

.6304 

6.827 

193 

-6383 

20.25 

6 

.6120 

.0576 

53 

.6174 

.4027 

100 

.6236 

i  .918 

147 

.6305 

7.001 

194 

.6385 

20.69 

7 

.6121 

.0603 

54 

.6175 

.4176 

101 

.6237 

i  .976 

148 

.6307 

7.178 

195 

.6387121.13 

8 

.6122 

.0630 

55 

.6177 

•  4331 

102 

.6238 

2.036 

149 

.6309 

7-359 

196 

.638921.58 

9 

.6123 

.0659 

56 

.6178 

•  4490 

103 

.6240 

2.098 

ISO 

.6310 

7-545 

197 

.639122.04 

10 

.6124 

.0690 

57 

•  6179 

.4055 

104 

.6241 

2.161 

151 

.6312 

7.736 

198 

.639322.50 

ii 

.6125 

.0722 

58 

.6180 

.4824 

105 

.6243 

2.226 

152 

.6313 

7.929 

199 

.639422.97 

12 

.6126 

.0754 

59 

.6182 

•  4999 

1  06 

.6244 

2  .294 

153 

.6315 

8.127 

2OO 

.639623.46 

13 

.6127 

.0789 

60 

.6183 

.5180 

107 

.6246 

2.362 

154 

.6317 

8.328 

2OI 

.639723.94 

14 

.6128 

.0824 

6l 

.6184 

.5367 

108 

.6247 

2.432 

155 

.6318 

8.534 

2O2 

.640024.44 

IS 

.6130 

.0862 

62 

.6185 

•  5559 

109 

.624812.504 

156 

.6320 

8.744 

203 

7640224.95 

16 

.6131 

.0900 

63 

.6187 

.5758 

no 

.6250-2.578 

157 

.6322 

8.958 

204 

.640425.4? 

134 


FAHRENHEIT  DEGREE  OF  TEMPERATURE  FROM  30°  BELOW  TO  434°  ABOVE  ZERO 


Temperature, 
Degrees 
Fahrenheit 

Values  of  K 

Values  of  H 

Temperature, 
Degrees 
Fahrenheit 

Values  of  K 

Values  of  H 

Temperature, 
Degrees 
Fahrenheit 

Values  of  K 

Values  of  H 

w 

Values  of  K 

Values  of  H 

Temperature, 
Degrees 
Fahrenheit 

Values  of  K 

Values  of  H 

205 

.6405;25-99 

251    .649961.89 

297 

.6607  130.8 

343 

.6736 

250.9 

389 

.6890444-4 

206 

.640726.53 

252    .650162.97 

298 

.6610132.8 

344 

.6739 

254-2 

390 

.6893449.6 

207 

.640927.07 

253 

.650364.08 

299 

.6612 

134-8 

345 

.6741 

257.6 

391 

.6897 

454-9 

208 

.6411  27.62 

254 

.650565.21 

300 

.6615 

136.8 

346 

.6745 

261.0 

392 

.6901 

460.2 

209 

.641328.18 

255 

.650866.34 

301 

.6617 

138.9 

347 

.6749 

264.5 

393 

.6905 

465.6 

210 

.641528.75 

256 

.651067.49 

302 

.6620 

141.0 

348 

.6751 

268.0 

394 

.6908 

470.9 

211 

.64i7;29.33 

257 

.651268.66 

303 

.6623 

I43-I 

349 

.6754 

271-5 

395 

.6911 

476.4 

212 

.641929.92 

258 

.651469.85 

304 

.6625 

145-3 

350 

.6757 

275.0 

396 

•  6915 

481.9 

213 

.642130.53 

259 

.6516 

71.05 

305 

.6628 

147.4 

351 

.6760 

278.6 

397, 

.6919 

487-4 

214 

.642331.14 

260 

.6518 

72.26 

306 

.6631 

149.6 

352 

.6763 

282.2 

398 

.6923 

493-0 

215 

.642431-76 

261 

.6521 

73.50 

307 

.6633 

151.8 

353 

.6767 

285.9 

399 

.6927 

498.7 

216 

.6426132.38 

262 

.6523 

74-75 

308 

.6636 

I54.I 

354 

.6770 

289.6 

400 

.6931 

504.4 

217 

.642833.02 

263 

.6525 

76.02 

309 

.6639 

156.3 

355 

.6773 

293-3 

401 

-6935 

SiO.i 

218 

.643033.67 

264 

.6528 

77.30 

310 

.6641 

158.7 

356 

.6776 

297-1 

402 

.6939 

515.9 

219 

.643234  -33 

265 

.6530 

78.61 

311 

.6644 

161.0 

357 

.6780 

300.9 

403 

.6943 

521.7 

22O 

.643435.01 

266 

.6532 

79-93 

312 

.6647 

163-3 

358 

.6783 

304.8 

404 

.6947 

527.6 

221    |.643635.69 

267 

.6534 

81.27 

313 

.6650165.7 

359 

.6786 

308.7 

405 

.6951 

533-5 

222      .643836.38 

268 

.6537 

82.62 

314 

.6652 

168.1 

360 

.6789 

312.6 

406 

.6955 

539.5 

223      .644037.08 

269 

.6539 

84.00 

315 

.6655 

170.5 

36l 

.6792 

316.6 

407 

.6958 

545.6 

224 

.644237.80 

270 

.6541 

85.39 

316 

.6658 

173-0 

362 

.6795 

320.6 

408 

.6962 

551.6 

225 

.6444 

38.53 

271 

.6543 

86.83 

317 

.6661 

175.5 

363 

.6799 

324-6 

409 

.6966 

557.8 

226 

.644639-27 

272 

.6546 

88.26 

3i8 

.6663 

178.0 

364 

.6803 

328.7 

410 

.6970 

564.0 

227 

.644840.02 

273 

-6548 

89.71 

319 

.6666 

180.6 

365 

.6806 

332.8 

411 

.6975 

570.2 

228 

.645140.78 

274 

.6551 

91.18 

320 

.6669 

183.1 

366 

.6809 

337-0 

412 

.6979 

576.5 

229 

.64S34I.S6 

275 

.6553 

92.67 

321 

.6671 

185.7 

367 

.6813 

341-2 

413 

.6983 

582.8 

230 

.645542.34 

276 

.6555 

94.18 

322 

.6674 

188.3 

368 

.6816 

354-4 

414 

.6987 

589.3 

231 

.645743.14 

277 

•655895.71 

323 

.6677191-0 

369 

.6820 

349-7 

415 

.6991 

595-7 

232 

.645843.95 

278 

.656097.26 

324 

.6680193.7 

370 

.6822 

354-0 

416 

.6995 

602.2 

233 

.646044.77 

279 

.6563 

98.83 

325 

.6683196.5 

371 

.6825 

358.4 

417 

.6999 

608.8 

234 

.6463 

4S.6i 

280 

-6565 

100.4 

326 

.6686 

199.2 

372 

.6829 

362.8 

418 

.7003 

615.4 

235 

.6465 

46.46 

281 

.6568 

102.0 

327 

.6689 

202.0 

373 

.6832 

367-3 

419 

.7007 

622.1 

236 

.6467 

47-32 

282 

.6570 

103.7 

328 

.6691 

204.8 

374 

.6836 

371.8 

420 

.7012 

628.8 

237 

.6469 

48-19 

283 

.6572 

105-3 

329 

.6694 

207.7 

375 

.6839 

376.3 

421 

.7016 

635.6 

238 

.6471 

49.08 

284 

•  6575 

I07-O 

330 

.6697 

210.5 

376 

.6843 

380.9 

422 

.7021 

642.5 

239 

.6473 

49.98 

285 

.6577 

108.7 

331 

.6700 

213-5 

377 

.6847 

385.5 

423 

.7025 

649.4 

240 

.6475 

50.89 

286 

.6580 

II0.4 

332 

.6703 

2l6.4 

378 

.6850 

390.2 

424 

.7029 

&6.  3 

241 

.6477 

51.83 

287 

.6582 

112.  1 

333 

.6707 

219-4 

379 

.6853 

394-9 

425 

•  7033 

663.3 

242 

.6479 

52.77 

288 

.6584 

II3-9 

334 

.6709 

222.4 

380 

.6857 

399-6 

426 

.7037 

670.4 

243 

.6481 

53-72 

289 

.6587 

US-  8 

335 

.6712 

225-4 

381 

.6861 

404-3 

427 

.7042 

677.5 

244 

.6484 

54-69 

290 

.6590 

H7-5 

336 

.6715 

228.5 

382 

.6865 

409-3 

428 

.7046 

684.7 

245 

.6486 

55.68 

291 

.6592 

II9-3 

337 

•  6717 

231.6 

383 

.6868 

414.2 

429 

.7051 

691.9 

246 

.648856.67 

292 

.6594 

121.  2 

338 

.6721 

234-7 

384 

.6871 

4I9.I 

430 

.7055 

699-2 

247 

.6490157.69 

293 

.6597 

I23.I 

339 

.6724 

237-9 

385 

.6875 

424.1 

431 

.7059 

706.5 

248 

.649258.71 

294 

.6600 

125.0 

340 

.6727 

24I.I 

386 

.6879 

429.1 

432 

.7064 

713.9 

249 

.6494 

59.76 

295 

.6602 

126.9 

341 

.6730 

244-3 

387 

.6882 

434.2 

433 

.7068 

721.4 

250    .649660.81 

296 

.6604 

128.8 

342 

.6733247.6 

388 

.6886439.3 

434 

.7073 

728.9 

135 


136 


COMPRESSED  AIR 


TABLE  IV.*— SPECIAL  TABLE  RELATING  TO  STAGE  COMPRESSION  FROM 

FREE  AIR  AT  14.7  POUNDS  PRESSURE  AND  62°  TEMPERATURE 

Compression  adiabatic  but  cooled  between  stages 


Single  Stage 

Two  Stage 

G 

£ 

.9 

3  k 

ill 

G 
O 

•S  ^3 

S°f 

i 

1 

%\ 

3 

I 

qj  0) 

Sfe 

£ 

1 

%  £   . 

• 

?  ^ 

f  d  & 

c  3 

g  « 

°  cS  o 

1 

0 

*'S 

1  v 

ft  "5> 

1  C  u 

J  ^ 

g  W  ^ 

fe  4)  ft 

o 

J>  *  « 

(C  3 

w  I 

H  § 

*!« 

"o  8 

£  ^  < 

1 

II  1 

|| 

l| 

§  P.  fa 

•2  a 

ill 

^  II 

$ 

a 

fo 

a 

(2 

I 

1 

Pf 

r 

W 

M.E.P. 

T, 

H.P. 

Vr 

r2 

H.P. 

5 

1-34 

.  IO20 

4.50 

108 

.0197 

10 

1.68 

.1279 

8.30 

144 

.0362 

15 

2.02 

•1537 

11.51 

177 

.0045 

20 

2.36 

.1796 

14.40 

207 

.0628 

25 

2.70 

.2055 

17.00 

235 

.0742 

30 

3-°4 

•2313 

19.40 

259 

•0845 

35 

3.38 

.2572 

21.65 

280 

.0944 

40 

3-72 

•2831 

23.60 

303 

.1030 

45 

4.06 

.3090 

25-50 

321 

.  III2 

5o 

4.40 

•3348 

27.50 

34i 

•H95 

2.  IO 

1  80 

.1063 

55 

4.74 

.3607 

29.10 

358 

.1268 

2.17 

189 

.1123 

00 

5-08 

.3866 

30.75 

373 

•1339 

2.25 

196 

.1184 

65 

5-42 

.4124 

32.30 

392 

.1408 

2.33 

200 

•1235 

70 

5.76 

.4383 

33.80 

405 

.1472 

2.40 

207 

.1286 

75 

6.10 

.4642 

420 

•1532 

2.47 

214 

.1329 

80 

6.44 

.4901 

36.55 

434 

.1590 

2.54 

222 

•1372 

85 

6.78 

•5T59 

37-90 

447 

•  ^50 

2.60 

227 

.1410 

90 

7.12 

.5418 

39.10 

461 

•1705 

2.67 

233 

.1462 

95 

7.46 

.5676 

40.35 

473 

•1758 

2-73 

238 

.1500 

IOO 

7.80 

•5935 

41.65 

485 

.1812 

2-79 

242 

.1542 

I05 

8.14 

.6194 

42.30 

497 

.1841 

2.85 

246 

•1578 

no 

8.48 

•6453 

43-75 

5o8 

.1908 

2.90 

251 

.1615 

"5 

8.82 

.6712 

45.16 

.1965 

2.99 

256 

.1648 

120 

9.16 

.6971 

46.00 

530 

.2008 

3.02 

259 

.1681 

125 

130 

9-50 
9.84 

.7230 
.7488 

47.05 
47.80 

540 
550 

.2045 
.2085 

3.08 
3-14 

262 
266 

.1710 
.1740 

135 

10.18 

•7747 

48.85 

•2135 

3-19 

269 

•1775 

140 

10.52 

.8005 

.49.90 

569 

.2176 

3-24 

272 

.1810 

!45 

10.86 

.8264 

51.00 

578 

.2220 

3-29 

276 

•1837 

11.20 

.8522 

5I-7° 

587 

•2255 

3-35 

280 

.1865 

*  The  table  is  limited  to  the  special  initial  condition  of  air  specified  in  the 
caption.  The  assumption  of  14.7  as  atmospheric  pressure  makes  the  weights 
and  work  a  little  in  excess  of  average  conditions.  However,  it  is  a  valuable  and 
very  instructive  table. 


TABLES 


137 


TABLE  IV '.—(Continued] 


Two  Stage 

Three  Stage 

a 

!S  o 

a 
o 

C     ' 

8°3 

c 

.0 

a  j. 

o^l 

.2 

3  w 

o  ^ 

$ 

u  fe 

U  [£  •- 

|£ 

V 

3 

a 
6 

S3 

£  & 
&J 

o  M 

it 

0    3  ^ 

O     fe 

4>      Q^ 

J3 
CO 

If 

°cj  fe 

I 

8 

«*H     ^_, 

O  •£ 

S  w  ^ 

£     £     '~ 

-g 

S  w  ^ 

^  r>  '~ 

£ 

o 

o 

O        Q 

J3    O 

o 

Sfl 

||| 

j- 

$& 

«   || 

S 

'5  &H 

S'S 

c  W  £ 

k  P,  fe 

*^3    c! 
rt   "^ 

c  W  ?i 

§  P,  £ 

o 

8 

* 

S 

W 

5 

S 

w 

Pg 

r 

w 

W1 

r2 

H.P. 

(r)S 

rs 

H.P. 

100 

7.8 

•5936 

2.79 

242 

•1542 

1.98 

176 

.1450 

150 

II.  2 

.8522 

3-35 

280 

.1865 

2.24 

200 

.1752 

2OO 

14.6 

I.IIIO 

3.82 

3o8 

.2110 

2-44 

215 

.1965 

250 

18.0 

1.3697 

4-24 

332 

•2315 

2.62 

226 

.2140 

300 

21.4 

1.6285 

4-63 

353 

.2490 

2.78 

241 

.2295 

350 

24.8 

1.8872 

4.98 

370 

.2640 

2.92 

251 

.2418 

4OO 

28.2 

2.1459 

5  31 

386 

.2770 

3-04 

259 

•2535 

450 

3i-6 

2  .  4048 

5-6i 

399 

.2895 

267 

.2630 

500 

2  .  6634 

5-91 

412 

•2915 

3-27 

275 

•2730 

550 

38.4 

2.9221 

3-37 

281 

.2830 

600 

41.8 

3.l8lO 

3-47 

287 

.2910 

650 

45-2 

3-4395 

3.56 

292 

.2960 

700 

48.6 

3.6982 

3-64 

297 

•3025 

75° 

52.0 

3-9570 

3-73 

302 

.3090 

800 

55-4 

4-2155 

307 

.3150 

850 

58.8 

4-4745 

3-83 

312 

.3210 

900 

62.2 

4-7330 

3-96 

316 

.3260 

95° 

65.6 

4.9920 

4-03 

320 

•3315 

IOOO 

69.0 

5-251° 

4.  10 

324 

•3360 

1050 

72.4 

5  •  5095 

4.17 

328 

.3400 

IIOO 

75-8 

5.7684 

4-23 

331 

•3445 

1150 

79.2 

6.0270 

4.29 

334 

•3490 

1200 

82.6 

6-2855 

4.36 

337 

•3525 

1250 

86.0 

6-5445 

4-4i 

34i 

•3570 

1300 

89.4 

6.8030 

4-47 

344 

3615 

1350 

92.8 

7.0620 

4-52 

347 

:366o 

I40O 

96.2 

7.3210 

4-58 

350 

.3685 

1450 

99.6 

7-5795 

4.64 

353 

.3710 

1500 

103.0 

7.8382 

4.70 

356 

•3740 

1550 

106.4 

8.0965 

4-75 

359 

.3780 

1600 

109.8 

8-355° 

4-79 

361 

•3820 

1650 

113.2 

8.6140 

4-83 

363 

•3850 

1700 

116.6 

8.8730 

4.87 

365 

.3880 

1750 

120.0 

9.1320 

4-93 

367 

•3915 

I800 

123-4 

9.3900 

4.97 

369 

•3940 

1850 

126.8 

9.6485 

5-02 

•3965 

138 


COMPRESSED  AIR 


TABLE  V.  —  VARYING  PRESSURES  WITH  ELEVATIONS 


Solution  of  formula  20,  Art.  17,  viz. 


=  1.  16866  -- 

1  22.4  1 


Elevation  in  Feet 

Pressure  in  Pounds  per  Square  Inch 

Temp.  50°  F. 

Temp.  35°F. 

Temp.  20°  F. 

0 

14.70 

14.70 

14  7°x 

IOOO 

14    17 

I4-I5 

14.14 

2000 

13.66 

l3-63 

13-99 

3000 

13.16 

13.12 

13.07 

4000 

12.69 

12.63 

12  57 

5000 

12.23 

12.  l6 

12  .09 

5280 

12.  10 

12.03 

II  .96 

6000 

11.78 

11.71 

11.63 

7000 

11.36 

11.27 

ii.  18 

8000 

10.95 

10.85 

JO-75 

9000 

10-55 

10.45 

10.33 

1  0000 

10.17 

10.  06 

9.94 

12500 

9.28 

9-15 

9  .02 

15000 

8.46 

8.32 

8.18 

TABLE  VI.* — HIGHEST  LIMIT  TO  EFFICIENCY  WHEN  COMPRESSED  AIR  is  USED 

WITHOUT  EXPANSION,  ASSUMING  ATMOSPHERIC  PRESSURE  =  14.5 

POUNDS  PER  SQUARE  INCH 


r 

h 

E 

r 

h 

E 

r 

k 

E 

I  .2 

6.66 

91.4 

S-2 

140.0 

49-o 

9.2 

273-3 

40.2 

1.4 

13-3 

84.9 

5-4 

146.6 

48.3 

9-4 

280.0 

39-9 

1.6 

20.  o 

79-8 

5-6 

153-3 

47-7 

9.6 

286.6 

39  -6 

1.8 

26.6 

75-6 

5-8 

1  60.0 

47-o 

9.8 

293-3 

39-3 

2.0 

33-3 

72.0 

6.0 

166.6 

46.5 

IO.O 

300.0 

39-o 

2.2 

40.0 

69.2 

6.2 

173-3 

46.0 

10.25 

308.3 

38.6 

2.4 

46.6 

66.7 

6-4 

180.0 

45-5 

10.50 

316.6 

38.5 

2.6 

53-3 

61.9 

6.6 

186.6 

45-° 

10.75 

325-o 

38.0 

2.8 

60.0 

62.4 

6.8 

193-3 

44-5 

11.00 

333-3 

37-9 

3-° 

66.6 

60.7 

7.0 

200.0 

44.0 

11.25 

341.6 

37-7 

3-2 

73-3 

59-i 

7-2 

206.6 

43-6 

II  .50 

35o-o 

37-4 

3-4 

80.0 

57-8 

7-4 

213-3 

43-i 

ii-75 

353-3 

37-i 

3-6 

86.6 

56.4 

7-6 

220.0 

42.8 

12.00 

366.6 

36-9 

3-8 

93-3 

55-2 

7-8 

226.6 

42.4 

12.25 

375-o 

36.7 

4.0 

IOO.O 

54-i 

8.0 

233-3 

42.0 

I2.5O 

383-3 

36-4 

4.2 

1  06.  6 

53-i 

8.2 

240.0 

41.7 

12-75 

391.6 

36.2 

4-4 

ii3-3 

52-i 

8.4 

246.6 

4i-4 

13.0 

400.0 

36.0 

4.6 

120.  O 

5i-3 

8.6 

253-3 

41.1 

14.0 

433-3 

35-2 

4.8 

126.6 

5°-5 

8.8 

26O.O 

40.8 

15.0 

466.6 

34-5 

5-o 

133-3 

49-7 

9.0 

266.6 

40.5 

16.0 

500.0 

33-8 

*  This  table  reveals  the  limit  of  efficiency  when  air  is  applied  without  utiliz- 
ing any  of  its  expansive  energy. 

The  column  headed  r  gives  the  ratio  of  compression,  while  that  headed  h  gives 
the  water  head  equivalent  to  a  pressure  given  by  the  ratio  r  on  the  assumption 
that  one  atmosphere  is  a  pressure  of  14.5  Ib.  per  square  inch  or  a  water  head  of 
33.3  ft.,  this  being  more  nearly  the  average  condition  than  14.7,  which  is  so 
commonly  taken. 

It  should  be  understood  that  this  efficiency  cannot  be  reached  in  practice — 
it  being  reduced  by  friction  of  air  and  machinery  and  by  clearance  in  any  form 
of  engine. 


TABLES 


139 


TABLE  VII. — EFFICIENCY  OF  DIRECT  HYDRAULIC  AIR  COMPRESSORS 

2.3  logio  r 
r  —  i 


Formula  33,  Art.  32,  viz.  E 


Water  Head 

Gage  Pressure 

Absolute  Pies- 
sure 

Atmospheres 

=  r 

Efficiencv, 
E 

0.0 

0.0 

14-5 

j 

I  .00 

33-3 

14-5 

29.0 

2 

.69 

66.6 

2Q.O 

43-5 

3 

•55 

100.  0 

43-5 

58.0 

4 

.46 

133-3 

58.0 

72-5 

5 

.40 

166.6 

72.5 

87.0 

6 

•36 

200.  o 

87.0 

101.5 

7 

•33 

233-3 

101.5 

116.0 

8 

•3° 

266.0 

116.0 

i3°-5 

9 

.28 

300.0 

130-5 

145.0 

10 

.26 

TABLE  VIII. — COEFFICIENT  "c"  FOR  VARIOUS  HEADS  AND  DIAMETERS 


d" 

*=i" 

i—  2" 

*'=3" 

t-4" 

t-s" 

A 

0.603 

0.606 

0.610 

0.613 

0.616 

i 

0.602 

0.605 

0.608 

0.610 

0.613 

I 

0.601 

0.603 

0.605 

0.606 

0.607 

li 

0.601 

0.601 

0.602 

0.603 

0.603 

2 

0.600 

0.600 

0.600 

0.600 

0.600 

2^ 

0-599 

0-599 

0-599 

0.598 

0.598 

3^ 

o  599 

0.598 

o-597 

0.596 

o  596 

S 

o-599 

o  597 

o  596 

0-595 

o  594 

4 

0.598 

o-597 

o-595 

0-594 

o-593 

4^ 

o  598 

0.596 

0.596 

0-593 

0.592 

Tables  VIII  and  Villa  give  the  experimental  coefficients  for  orifices  for  deter- 
mining the  weight  of  air  passing  by  formula: 

For  round  orifices  Weight  (Q]  =  c  0.1639 d2 -\/- p 

For  rectangular  orifices         Weight  (Q)  =  c  2.413  0\/-  p 

Q  =  Weight  of  air  passing  in  pounds  per  second. 
c  =  Experimental  coefficient. 
d  =  Diameter  of  orifice  in  inches. 

i  =  Difference  of  pressure  inside  and  outside  of  orifice  in  inches  of  water. 
/  =  Absolute  temperature  of  air  back  of  orifice. 
a  =  Area  of  rectangular  orifice  in  square  feet. 

p  =  Absolute  pressure  back  of  orifice  in  pounds  per  square  inch  =  atmospheric 
pressure  -j-  0.036  i. 


140 


COMPRESSED  AIR 
TABLE  Villa 


Coefficients  "c"  for  Large  Orifices 

McGill 

Water 

Coeffi- 

Gage 
]  Inches 

cient 
Orifice 

Round 

Square 

30" 

24" 

1  8" 

24"X24" 

i8"Xi8" 

i8"X3<>" 

I 

•599 

.604 

•599 

•597 

.607 

•598 

.602 

2 

•597 

.602 

•597 

.596 

.605 

.596 

.600 

3 

•596 

.601 

•596 

•594 

.604 

•595 

•  599 

4 

•597 

.000 

•595 

•593 

.603 

.694 

•  598 

5 

•594 

•599 

•594 

•592 

.601 

•593 

•  597 

From  Table  IX  can  be  readily  found  friction  losses  in  air  pipes  as  computed 
by  the  author's  formula:  Art.  (26),  viz., 


The  table  conforms  to  values  of  c  taken  from  the  curve  A,  B,  Plate  II,  using  the 
National  Tube  Works  standard  for  actual  diameters  as  shown  here. 

NATIONAL  TUBE  WORKS  STANDARD  WITH  COEFFICIENTS  FOR  FORMULA  (26) 


Nominal  diameters.  .  . 

H 

X 

I 

IK 

IK 

iK 

2 

*K 

3 

Actual  diameters  .... 

.666 

.824 

1.048 

1.380 

1.610 

1.820 

2.067 

2.468 

3-067 

Coefficient  

.170 

.107 

.099 

.091 

.087 

.084 

.O8l 

0.765 

•0715 

Nominal  diameters  .  .  . 

31A 

4 

4^ 

5 

6 

8 

10 

12 

Actual  diameters  

3.548 

4.067 

4-508 

5-045 

6.065 

7.981 

10.018 

12.00 

Coefficient  

.0685 

.0660 

•0635 

.0615 

.0580 

•0535 

.0500 

.0480 

TABLES 
TABLE  IX. — FRICTION  IN  AIR  PIPES 


141 


Cubic  Feet 
Free  Air 
per  Minute 

Divide  the  Number  Corresponding  to  the  Diameter  and  Volume  by  the 
Ratio  of  Compression.     The  Result  is  the  Loss  in  Pounds  per 
Square  Inch  in  1000  feet  of  Pipe 

Nominal  Diameter  in  Inches 

H 

3/4 

i 

iH 

iH 

iN 

2 

2K 

3 

5 

10 

IS 
20 

25 

30 
35 
40 
45 
50 

60 
70 
80 
90 

IOO 

no 
1  20 

130 
140 

150 

160 
170 
180 
190 
200 

220 
240 
26O 
280 
300 

12.7 

50-7 
114.1 

1.2 

7.8 
17.6 

30-4 
50.0 

70.4 
95-9 
125.3 

2.2 
4.9 

8.7 
13-6 

19.6 
26.6 
34-8 
44.0 

54-4 

78.3 
106.6 
139.2 

2.O 
3-2 

4-5 

6.2 

8.1 

10.2 
12.0 

18.2 

24.7 

32.3 
40.9 

50.5 

61.1 
72.7 
85.3 
98.9 

113.6 
129.3 

2.7 
3-6 
4-5 
5-6 

8.0 
10.9 

14-3 
18.1 
22.3 

27.0 
32.2 
37-8 
43-8 

50.3 
57-2 
64.6 
72.6 
80.7 
89.4 

108.2 
128.7 

1.9 
2.4 
2.9 

4-2 
5-7 
7-5 
9-5 
11.7 

14.1 
16.8 
19.7 
22.9 

26.3 
29.9 
33-7 
37-9 
42.2 

46.7 

56.5 
67.3 
79.0 
91.6 
105.1 

2.2 
2.9 

3-8 
4.8 
6.0 

7.2 
8.6 

10.  1 

«.  7 

13-4 
15.3 
17.6 
19.4 
21.5 
23-9 

28.9 
34-4 
40.3 
46.8 

53-7 

2.8 

3-3 
3-9 
4.6 

5.2 
5-9 
6-7 
7-5' 
8.4 
9-3 

II-3 
13-4 
15-7 
18.2 
20.9 

2.1 
2-9 

3-5 
4.2 
4.9 

5-7 
6.6 

:::: 

:::: 

.... 

.... 

142 


COMPRESSED  AIR 
TABLE  IX. — (Continued) 


Nominal  Diameter  in  Inches 

2 

2* 

3 

3H 

4 

& 

5 

6 

8 

320 

61.1 

23-8 

7-5 

3-5 

34<> 

69.0 

26.8 

8-4 

3-9 

360 

77-3 

30.1 

9-5 

4.4 

38o 

86.1 

33-5 

10.5 

5-9 

400 

94-7 

37-1 

ii.  7 

5-4 

2.7 

420 

105.2 

40.9 

12.9 

6.0 

3.1 

440 

HS.5 

44-9 

14.1 

6.6 

3.4 

460 

125.6 

48.8 

15-4 

7-1 

3.7 

480 

.... 

53-4 

16.8 

7.8 

4.0 

500 

.... 

58.0 

18.3 

8.5 

4-3 

525 

.... 

64.2 

20.2 

9-4 

4-8 

2.6 

550 

.... 

70.2 

22.1 

10.2 

5-2 

2.9 

575 

.... 

76.7 

24.2 

II.  2 

5-7 

3.1 

600 

.... 

83.5 

26.3 

12.2 

6.2 

3.4 

625 

92.7 

28.5 

13-2 

6.8 

3.7 

650 

.... 

98.0 

39-9 

14-3 

7-3 

4.0 

675 

105.7 

33-3 

15-4 

7-9 

4.3 

700 

.... 

II3-7 

35-8 

16.6 

8.5 

4.6 

750 

.... 

130.5 

41.1 

I9.O 

9-7 

5-3 

2.9 

800 

46.7 

21-7 

ii  .  i 

6.1 

3-2 

*TW        / 

52.8 

24.4 

12  .  5 

6.8 

•  «J 

3-8 

o    * 
en   i 

27    4 

14  .  o 

7    7 

42 

o  <;o 

oy  •  * 

^  /    "  T" 

15.  7 

/  *  / 

8.6 

•  * 
47 

IOOO 

.... 

73-0 

33-8 

17.3 

9.5 

•  / 

5-2 

10^0 

80.5 

27.  7 

19.  i 

IO   4 

5-8 

IIOO 

88.4 

O  /     O 

40.9 

21.0 

AW  .  £f. 
H.5 

6-3 

2.4 

JJt-Q 

96.6 

44-7 

22  .9 

12.5 

6.9 

2.6 

I2OO 

.... 

.... 

V 

105.2 

48.8 

*    •  y 

25.0 

13-7 

\j  .  \j 

7-5 

2.8 

1300 

.... 

.... 

123.4 

57-2 

29.3 

16.0 

8.8 

3-3 

1400 

66.3 

•22  .Q 

18.6 

IO.  2 

3-8 

*^p*%p 

O 

76.1 

oo  y 

21  -3 

ii.  8 

O 

4.4 

1600 

1  \f  •  J. 

86.6 

44    3 

24.  2 

13.4 

5.  i 

1700 

.... 

97-8 

fT"  *  O 
50.1 

27.4 

5-7 

1800 

IIO.O 

56.1 

30.7 

16.9 

6.4 

TABLES 

TABLE  IX. — (Continued} 


143 


Nominal  Diameter  in  Inches 

4 

4W 

5 

6 

8 

10 

12 

1900 

2OOO 
2IOO 
22OO 
2300 

24OO 
2500 
2600 
2700 
2800 

2QOO 
3000 
3200 
3400 
3600 

3800 
4OOO 
42OO 
4400 
4600 

4800 
5000 
5250 
5500 
5750 

6OOO 
6500 
7OOO 
7500 
8000 

•   9OOO 
IOOOO 
IIOOO 
I2OOO 
13000 

I4OOO 
15000 
16000 
18000 
20000 

22000 
24000 
260CO 
28000 
30000 

62.7 
69-3 
76.4 
83.6 

91  .6 

99.8 
108.3 
117.2 

34-2 
37-9 
40.8 
45-8 
50.1 

54-6 
59-2 
64.0 
69-1 
74-3 

79.8 
85-2 

97-i 

109.5 

122.8 

18.9 
21.3 
23.0 

25-3 
27.6 

30.1 
32.6 
35-3 
38.1 
41.0 

43-9 

47-o 
53-5 
60.4 
67.7 

75-5 
83-6 
92.1 

IOI  .  2 
110-5 

120-4 

7-i 

7.8 

8-7 
9-5 
10.4 

ii.  3 
12.3 

13-3 
14-3 
iS-4 

16.5 

17.7 

20.1 
22.7 
25-4 

28.4 
31-4 

34-6 
38.1 
4i-5 

45-2 
49-1 
54-i 
59-4 
64.9 

70.7 
82.9 

96.  2 
II0.5 
125-7 

2.0 
2.2 
2-4 

2.6 
2.9 

3-i 
3-3 

3.6 

3-9 
4-i 
4-7 
5-3 
5-6 

6.6 

7-3 
8.1 
8.9 
9-7 

io-5 
ii-5 

12.6 

13-9 
15.2 

16.5 
19.8 
22.5 
25-8 
29.4 

37-2 
45-9 
55-5 
66.1 

77-5 

89.9 
103.2 
117.7 
148.7 

2.9 

3-2 

3-4 
3-8 

4-2 

4-6 

5-0 

5-9 
6.8 

7-8 
8.8 

II.  2 

13-8 
16.7 
19.8 
23-3 

27.0 
31-0 

35-3 
44-6 
55-0 

66.9 
79-3 
93-3 
108.0 

123-9 

% 

2-3 
2.6 

3-o 
3-6 

4-4 
5-4 
6.5 
7-7 
9.0 

10.5 

12.0 
13.7 
17-4 
21.4 

26.0 
30.1 
36.3 
42.1 
48.2 

.... 

.... 

.... 

144 


COMPRESSED  AIR 


TABLE  X. — TABLE  or  CONTENTS  OF  PIPES  IN  CUBIC  FEET  AND  IN  U.  S.  GALLON 


Diam. 

For  i  Foot 

in  Length 

Diam. 

For  i  Foot 

in  Length 

Diam. 
in 
nches 

in  Deci- 
mals of 
a  Foot 

Cubic  Feet. 
Also  Area 
in  Square 
Feet 

Gallons  of 
231  Cubic 
Inches 

Diam. 
in 
Inches 

in  Deci- 
mals of 
a  Foot 

Cubic  Feet. 
Also  Area 
in  Square 
Feet 

Gallons  of 
231  Cubic 
Inches 

I 

.0208 

.0003 

.0026 

II. 

.9167 

.6600 

4-937 

A 

.0260 

.0005 

.0040 

i 

•9375 

.6903 

5-l63 

1 

•0313 

.0008 

.0057 

i 

.9583 

•7213 

5-395 

1 

•°365 
.0417 

.0010 
.0014 

.0078 

.0102 

! 

12. 

•9792 
Foot 

•7530 
•7854 

5-633 
5.876 

A 

.0469 

.0017 

.0129 

i 

.042 

•8523 

6-375 

1 

.0521 

.0021 

.0159 

13" 

-083 

.9218 

6.895 

tt 

•0573 

.0026 

.0193 

\ 

.125 

•9940 

7-435 

2 

.0625 

.0031 

.0230 

14. 

.167 

.069 

7-997 

H 

.0677 

.0036 

.0270 

\ 

.208 

.147 

8-578 

I 

.0729 

.0042 

.0312 

*$''• 

.250 

.227 

9.180 

.  tt 
i. 

.0781 
•0833 

.0048 
•0055 

•0359 
.0408 

16. 

.292 
•333 

.310 
•396 

9.801 
10.44 

i 

.1042 

.0085 

.0638 

i 

•375 

-485 

ii  .11 

i 

.1250 

.0123 

.0918 

17- 

.417 

•576 

11.79 

I 

.1458 

.0168 

.1250 

i 

-458 

.670 

12.50 

2. 

.1667 

.0218 

.1632 

18. 

.500 

.767 

13.22 

J 

.1875 

.0276 

.2066 

\ 

•  542 

.867 

13.97 

| 

.2083 

.0341 

•255° 

19. 

-583 

.969 

14-73 

1 

.2292 

.0413 

•3085 

\ 

.625 

•074 

15-52 

3- 

.2500 

.0491 

•3673 

20. 

.666 

2.182 

16.32 

.2708 

.0576 

.4310 

\ 

.708 

2.292 

17-15 

| 

.2917 

.0668 

.4998 

21. 

•750 

2.405 

17.99 

| 

•3I25 

.0767 

•5738 

\ 

.792 

2.521 

18.86 

4- 

•3333 

•0873 

.6528 

22. 

•833 

2.640 

19-75 

i 

•3542 

•0985 

•737° 

* 

•875 

2.761 

20.65 

i 

•3750 

.1105 

•  8263 

23- 

.917 

2.885 

22.58 

I 

.3958 

.1231 

.9205 

\ 

.958 

3-012 

2i-53 

5- 

.4167 

.1364 

i.  020 

24. 

2.OOO 

3.142 

23-5° 

•4375 

•I5°3 

1.124 

25- 

2.083 

3-409 

25-50 

} 

•4583 

.1650 

1-234 

26. 

2.166 

3.687 

27-58 

i 

•4792 

.1803 

1-349 

27. 

2.250 

3-976 

29-74 

6. 

.5000 

.1963 

1.469 

28. 

2-333 

4.276 

31-99 

i 

•  5208 

.2130 

1-594 

29. 

2.416 

4.587 

34-31 

1 

•5417 

•2305 

1.724 

3°- 

2.500 

4.909 

36.72 

i 

•5625 

.2485 

1.859 

3i- 

2-583 

5.241 

39.21 

7 

•5833 

.2673 

1.999 

32. 

2.666 

5-585 

41-78 

i 

.6042 

.2868 

2.144 

33- 

2.750 

5-940 

44-43 

.6250 

.3068 

2.295 

34 

2-833 

6.305 

47-17 

| 

•  6458 

•3275 

2.450 

35- 

2.916 

6.681 

49.98 

8. 

.6667 

•3490 

2.611 

36. 

3.000 

7.069 

52.88 

i 

.6875 

•3713 

2.777 

37. 

3-083 

7.468 

55-86 

•7083 

•3940 

2.948 

38. 

3.166 

7.876 

58.92 

| 

.7292 

•4175 

3-125 

39- 

3-250 

8.296 

62.06 

9- 

.7500 

.4418 

3-3°5 

40. 

3-333 

8.728 

65.29 

.7708 

.4668 

3-492 

41. 

3.416 

9.168 

68.58 

1 

.7917 

•4923 

3.682 

42. 

3-5oo 

9  .  620 

71.96 

•  8125 

•5i85 

3-879 

43- 

3-583 

10.084 

75-43 

10. 

•8333 

•5455 

4.081 

44. 

3.666 

10.560 

79.00 

i 

.8542 

•573° 

4.286 

45- 

3-75° 

i  i  .  044 

82.62 

.8750 

.6013 

4.498 

46. 

3-833 

11.540 

86.32 

| 

.8958 

•  6303 

4-7*4 

47- 

3.916 

12.048 

90.  12 

48. 

•  4  .  ooo 

12.566 

94.02 

TABLE  XI. — CYLINDRICAL  VESSELS,  TANKS,  CISTERNS,  ETC. 

Diameter  in  Feet  and  Inches,  Area  in  Square  Feet,  and  U.  S.  Gallons  Capacity 

for  One  Foot  in  Depth 

i  cu.  ft. 

i  gal.  =  231  cu.  in.  = —  =  0.13368  cu.  ft. 

7.4505 


Diam. 

Area 

Gals. 

Diam. 

Area 

Gals. 

Diam. 

Area 

Gals. 

Ft.    In. 

Sq.  Ft. 

i  Ft. 
Depth 

Ft.    In. 

Sq.  Ft. 

i  Ft. 
Depth 

Ft.    In. 

Sq.  Ft 

i  Ft. 
Depth 

.785 

5.89 

5      5 

23.04 

172.38 

I7      6 

240.53 

1799.3 

I 

.922 

6.89 

5     6 

23.76 

I77-72 

17    9 

247-45 

1851.1 

2 

.069 

8.00 

5     7 

24.48 

183-15 

18 

254-47 

1903.6 

3 

.227 

9.18 

5     8 

25.22 

188.66 

18    3 

261.59 

1956.8 

4 

.396 

10.44 

5     9 

25-97 

194.25 

18     6 

268.80 

2010.8 

5 

.576 

11.79 

5  10 

26.73 

199.92 

18     9 

276.12 

2065.5 

6 

.767 

13.22 

5  ii 

27.49 

205.67 

19 

283.53 

2120.9 

7 

.969 

14-73 

6 

28.27 

211.51 

i9    3 

291.04 

2177.1 

8 

2.182 

16.32 

6    3 

30.68 

229.50 

19     6 

298.65 

2234.0 

9 

2.405 

17.99 

6     6 

33-18 

248.23 

19     9 

306.35 

2291.7 

10 

2.640 

19-75 

6    9 

35.78 

267.69 

20 

3U.l6 

2350.1 

ii 

2.885 

21-58 

7 

38.48 

287.88 

20     3 

322.06 

2409.2 

2 

3.142 

23.5° 

7    3 

41.28 

308.81 

20       6 

330.06 

2469.1 

2       I 

3-409 

25-5o 

7     6 

44-18 

330-48 

20     9 

338.16 

2529.6 

2       2 

3.687 

27-58 

7    9 

47-17 

352-88 

21 

346.36 

2591.0 

2     3 

3-976 

29.74 

8 

50.27 

376.01 

21     3 

354-66 

2653.0 

2       4 

4.276 

31-99 

8    3 

53-46 

399-88 

21       6 

363-05 

2715.8 

2     5 

4.587 

34.31 

8     6 

56.75 

424.48 

21       9 

371-54 

2779.3 

2       6 

4.909 

36-72 

8    9 

60.13 

449-82 

22 

380.13 

2843.6 

2     7 

5-241 

39-21 

9 

63.62 

475-89 

22       3 

388.82 

2908.6 

2       8 

5-585 

41-78 

9     3 

67.20 

502.70 

22       6 

397-6i 

2974.3 

2     9 

5-940 

44.43 

9     6 

70.88 

53o-24 

22       9 

406.49 

3040.8 

2    10 

6.305 

47.16 

9     9 

74.66 

558.51 

23 

4I5-48 

3108.0 

2     II 

6.681 

49.98 

10 

78.54 

587-52 

23       3 

424.56 

3!75-9 

3 

7.069 

52.88 

10     3 

82.52 

617.26 

23     6 

433-74 

3244.6 

3     i 

7.467 

55-86 

10     6 

86.59 

647.74 

23    9 

443-01 

33i4.o 

3     2 

7.876 

58.92 

10     9 

90.76 

678.95 

24 

3384-1 

3     3 

8.296 

62.06 

ii 

95.03 

710.90 

24    3 

461  .86 

3455-0 

3     4 

8.727 

65.28 

ii     3 

99.40 

743-58 

24     6 

471-44 

3526.6 

3     5 

9.168 

68.58 

ii     6 

103-87 

776.99 

24    9 

481.11 

3598.9 

3     6 

9.621 

7i-97 

ii     9 

108.43 

811.14 

25 

490.87 

3672.0 

3     7 

10.085 

75-44 

12 

113.10 

846.03 

25     3 

500.74 

3745-8 

3     8 

10-559 

78.99 

12     3 

117.86 

881  .  65 

25     6 

510.71 

3820.3 

3     9 

11.045 

82.62 

12       6 

122.72 

918.00 

25    9 

520.77 

3895-6 

3  10 

11-541 

86.33 

12     9 

127.68 

955-09 

26 

530-93 

3971-6 

3  ii 

12.048 

90.13 

13 

132.73 

992-91 

26    3 

54I-I9 

4048.4 

4 

12.566 

94.00 

13     3 

137-89 

1031-5 

26     6 

551-55 

4125-9 

4     i 

13-095 

97.96 

i3     6 

I43-I4 

1070.8 

26    9 

562.00 

4204.1 

4     2 

13-635 

102.00 

i3     9 

148.49 

iiio.S 

27 

572.56 

4283.0 

4     3 

14.186 

106.12 

i4 

153-94 

ii5i-5 

27     3 

583-21 

4362.7 

4     4 

14.748 

110.32 

14     3 

159.48 

1193.0 

27     6 

593-96 

4443-1 

4     5 

15-321 

114.61 

14     6 

165-13 

1235-3 

27    9 

604.81 

4524.3 

4     6 

I5-90 

118.97 

14    9 

170.87 

1278.2 

28 

6i5.75 

4606.2 

4     7 

16.50 

123.42 

i5 

176.71 

1321.9 

28    3 

626.80 

4688.8 

4     8 

17.10 

127-95 

i5     3 

182.65 

1366.4 

28     6 

637-94 

4772-i 

4     9 

17.72 

132.56 

15     6 

188.69 

1411.5 

28     9 

649  .  18 

4856.2 

4  10 

i8.35 

137-25 

T5     9 

194.83 

1457-4 

29 

660.52 

4941.0 

4  ii 

18.99 

142.02 

16 

201.06 

1504.1 

29    3 

671.96 

5026.6 

5 

19.63 

146.88 

16     3 

207.39 

I55L4 

29     6 

683.49 

5112.9 

5     i 

20.29 

151.82 

16    6 

213.82 

1599-5 

29     9 

695-13 

5I99-9 

5     2 

70.97 

156.83 

16    9 

220.35 

1648.4 

3° 

706.86 

5287.7 

5    3' 

21.65 

161.93 

i7 

226.98 

1697.9 

5     4 

22.34 

167.12! 

i7     3 

233  •  7  J 

1748.2 

10 


145 


146 


COMPRESSED  AIR 


TABLE  XII. — STANDARD  DIMENSIONS  OF  WROUGHT-IRON  WELDED  PIPE 
(National  Tube  Works) 


Nominal 
Inside 
Diameter 

Actual 
Outside 
Diameter 

Actual 
Inside 
Diameter 

Internal  Area 

External  Area 

Ins. 

Ins. 

Ins. 

Sq.  In. 

Sq.  Ft. 

Sq.  In. 

Sq.  Ft. 

i 

•405 

.270 

•057 

.0004 

.1288 

.0009 

i 

•540 

.364 

.104 

.0007 

.2290 

.0016 

i 

•675 

•493 

.191 

.0013 

•3578 

.0025 

\ 

.840 

.622 

•3°4 

.0021 

•554 

.0038 

i 

1.050 

.824 

•533 

.0037 

.866 

.0060 

I.3I5 

1.048 

.861 

.0060 

1.358 

.0094 

ij 

1.  660 

1-380 

1.496 

.OI04 

2.  164 

.0150 

i| 

1.900 

1.610 

2.036 

.OI4I 

2.835 

.0197 

2 

2-375 

2.067 

3-356 

•0233 

4-43° 

.0308 

2i 

2-875 

2.468 

4.780 

•0332 

6.492 

.0451 

3 

3-5°o 

3-067 

7-383 

9.621 

.0668 

ai 

4.000 

3.548 

9.887 

!o689 

12.566 

.0875 

4 

4.500 

4.026 

12.730 

.0884 

15.904 

.1104 

4* 

5.000 

4.508 

15.961 

.1108 

19.635 

.1364 

5 

5.563 

5.045 

19.986 

.1388 

24.301 

.1688 

6 

6.625 

6.065 

28.890 

.2006 

34.472 

.2394 

7 

7.625 

7.023 

38.738 

.2690 

45  •  664 

•3171 

8 

8.625 

7-981 

50.027 

•3474 

58.426 

•  4057 

9 

9.625 

8-937 

62.730 

•4356 

72.760 

•5053 

10 

10.75 

10.018 

78.823 

•5474 

90.763 

•  6303 

ii 

n-75 

I  I  .  000 

95-033 

.6600 

108.434 

•  7530 

12 

12-75 

I2.OOO 

113.098 

.7854 

127.677 

.8867 

13 

14 

I3.25 

137.887 

•9577 

153-938 

1.0690 

14 

15 

14.25 

159-485 

1.1075 

176.715 

1.2272 

15 

16 

15.25 

182.665 

1.2685 

201.062 

1-3963 

17 

18 

17-25 

239.706 

1.6229 

254.470 

1.7671 

19 

20 

I9-25 

291.040 

2.  O2  I  I 

314.159 

2.1817 

21 

22 

21.25 

354.657 

2.4629 

380.134 

2.6398 

23 

24 

424.558 

2.9483 

452.390 

3.1416 

TABLES 


147 


TABLE  XIII. — HYPERBOLIC  LOGARITHMS 


N. 

Loga- 
rithm. 

N. 

Loga- 
rithm. 

N. 

Loga- 
rithm. 

N. 

Loga- 
rithm. 

.01 

.00995 

1.57 

.45108 

2.13 

.75612 

2.69 

.98954 

.02 

.01980 

1.58 

•45742 

2.14 

.76081 

2.70 

.99325 

•03 

.02956 

1.59 

•46373 

2.15 

•76547 

2.71 

.99695 

.04 

.03922 

1.  60 

.47000 

2.l6 

.77011 

2.72 

.00063 

•05 

.04879 

1.61 

.47623 

2.17 

•77473 

2.73 

.00430 

.06 

•05827 

1.62 

•48243 

2.l8 

•77932 

2.74 

.00796 

.07 

.06766 

1.63 

.48858 

2.19 

•78390 

2-75 

.01160 

.08 

.07696 

1.64 

.49470 

2.20 

.78846 

2.76 

•01523 

.09 

.08618 

1.65 

.50078 

2.21 

.79299 

2.77 

.01885 

.10 

•09531 

1.66 

.50681 

2.22 

•79751 

2.78 

.02245 

.11 
.12 

•  10436 
•"333 

1.67 
1.68 

.51282 
•5l879 

2.23 
2.24 

.80200 
.80648 

I'R 

.02604 
.02962 

•13 

.12222 

1.69 

.52473 

2.25 

.81093 

2.81 

•03318 

.14 

•I3I03 

1.70 

•53063 

2.26 

.81536 

2.82 

.03674 

•13977 

1.71 

.53649 

2.27 

.81978 

2.83 

.04028 

.l6 

.  14842 

1.72 

•54232 

2.28 

.82418 

2.84 

.  04380 

.17 

.15700 

1-73 

.54812 

2.29 

.82855 

2.85 

•04732 

.l8 

i.74 

.55389 

2.30 

•83291 

2.86 

.05082 

.19 

•17395 

1-75 

.55962 

2.31 

•83725 

2.87 

•05431 

.20 

.18232 

1.76 

•56531 

2.32 

•84157 

2.88 

•05779 

.21 

.  19062 

.57098 

2-33 

.84587 

2.89 

.06126 

.22 

.19885 

1.78 

.57661 

2-34 

.85015 

2.90 

.06471 

•23 

.24 

.2O7OI 
.21511 

l:£ 

.58222 
•58779 

2-35 
2.36 

.85442 
.85866 

2.91 
2.92 

.06815 
.07158 

:li 

.22314 
.23111 

1.81 
1.82 

•59333 
.59884 

2-37 
2.38 

.86289 
.86710 

2.93 
2-94 

.07500 
.07841 

.27 

.  23902 

1.83 

.60432 

2-39 

.87129 

2.95 

.08181 

.28 

.  24686 

1.84 

.60977 

2.40 

•87547 

2.96 

.08519 

.29 

•25464 

1.85 

.61519 

2.4I 

.87963 

2.97 

..08856 

•3° 

.26236 

1.86 

.62058 

2.42 

•88377 

2.98 

.09192 

•31 

.27003 

1.87 

•62594 

2-43 

.88789 

2.99 

.09527 

•32 

.27763 

.88 

•63127 

2-44 

.89200 

3.00 

.09861 

•33 

.28518 

.89 

•  63658 

2-45 

.89609 

3.01 

.10194 

•34 

.29267 

.90 

.64185 

2.46 

.90016 

3.02 

.10526 

•35 

.30010 

.91 

.64710 

2-47 

.90422 

3.03 

.  10856 

•36 

.30748 

.92 

•65233 

2.48 

.90826 

3.04 

.11186 

•37 

.31481 

•93 

•65752 

2.49 

.91228 

3.05 

.11514 

.38 

.32208 

•94 

.66269 

2.50 

.91629 

3.06 

.  11841 

1-39 

.32930 

i.95 

.66783 

2.51 

.92028 

3-07 

.12168 

1.40 

•33647 

1.96 

.67294 

2.52 

.92426 

3.08 

•12493 

1.41 

•34359 

1.97 

•  67803 

2.53 

.92822 

3-09 

.12817 

1.42 

.35066 

1.98 

.68310 

2.54 

.93216 

3.10 

.13140 

1.43 

•35767 

1.99 

.68813 

2.55 

.93609 

.13462 

1.44 

•  36464 

2.00 

•69315 

2.56 

.94001 

3.12 

•13783 

1.45 

•37156 

2.01 

.69813 

2-57 

•94391 

3-13 

.14103 

1.46 

•37844 

2.02 

.70310 

2.58 

•94779 

3.14 

.14422 

1.47 

•38526 

2.03 

.  70804 

2.59 

.95166 

3-15 

•  14740 

1.48 

.39204 

2.04 

•71295 

2.60 

•95551 

3.16 

•15057 

1.49 

.39878 

2.05 

.71784 

2.61 

•95935 

3.17 

•15373 

.50 

•40547 

2.06 

.72271 

2.62 

•96317 

3.18 

.15688 

•Si 

.41211 

2.07 

•72755 

2.63 

.96698 

3-19 

.16002 

•52 

.41871 

2.08 

•73237 

2.64 

•97078 

3.20 

•16315 

•53 

•42527 

2.09 

•737i6 

2.65 

•97454 

3.21 

.16627 

•54 

.43178 

2.10 

.74194 

2.66 

•97833 

3.22 

.16938 

•55 

•43825 

2.  II 

.74669 

2.67 

.98208 

3.23 

.17248 

.56 

.44460 

2.12 

•75142 

2.68 

.98582 

3-24 

•17557 

148 


COMPRESSED  AIR 


TABLE  XIII.    Continued. — HYPERBOLIC  LOGARITHMS 


N. 

Loga- 
rithm. 

N. 

Loga- 
rithm. 

N. 

Loga- 
rithm. 

N. 

Loga- 
rithm. 

3-25 
3.26 

17865 
.18173 

3.8i 
3.82 

L33763 
1.34025 

4-37 
4.38 

1.47476 
I-4770S 

4-93 
4-94 

•59534 
•59737 

3-27 

.18479 

3.83 

1.34286 

4-39 

1-47933 

4-95 

•59939 

3.28 

.  18784 

3-84 

1-34547 

4.40 

1.48160 

4.96 

.60141 

3.29 

19089 

3-85 

1.34807 

4.41 

1.48387 

4-97 

.  60342 

3-30 

•  19392 

3-86 

1-35067 

4.42 

1.48614 

4.98 

•  60543 

3-31 

.19695 

3.87 

I-35325 

4-43 

i  .  48840 

4-99 

.60744 

3-32 

.  19996 

3.88 

I-35584 

4-44 

i  .  49065 

5-00 

.60944 

3-33 

.20297 

3.89 

i-3584i 

4-45 

i  .  49290 

5-01 

.61144 

3-34 

.20597 

3.90 

1.36098 

4.46 

*•  495  i5 

5-02 

-61343 

3-35 
3.36 

.  20896 
.21194 

3.9i 
3.92 

I.36354 
i  .  36609 

4-47 
4.48 

1-49739 
i  .  49962 

5.03 
5.04 

.61542 
.61741 

3-37 

.21491 

3-93 

1.36864 

4-49 

1.50185 

5-05 

.61939 

7  \J  7 

3.38 

.21788 

3-94 

1.37118 

4.50 

i  .  50408 

5.o6 

•62137 

3-39 

.22083 

3-95 

I-3737I 

4.51 

i  .  50630 

5.07 

•62334 

340 

.22378 

3.96 

1.37624 

4.52 

i  .  5085  i 

5.08 

•62531 

3.4i 

.22671 

3-97 

I-37877 

4-53 

1.51072 

5.09 

.62728 

342 

.22964 

3.98 

1.38128 

4-54 

I-S"93 

5.10 

.62924 

3-43 

.23256 

3-99 

I.38379 

4.55 

L5I5I3 

5.11 

.63120 

3-44 

•23547 

4.00 

i  .  38629 

4.56 

1-51732 

5.12 

•63315 

3-45 

•23837 

4.01 

1.38879 

4-57 

i-5i95i 

5.13 

•635II 

346 

1.24127 

4.02 

1.39128 

4.58 

1.52170 

5.14 

•63705 

3-47 

1.24415 

4.03 

1-39377 

4-59 

1.52388 

5.15 

.  63900 

348 

1.24703 

4.04 

1.39624 

4.60 

1.52606 

5.16 

.-64094 

3-49 

.  24990 

4-05 

1.39872 

4.61 

1.52823 

5.17 

.64287 

3-50 

.25276 

4.06 

1.40118 

4.62 

I-53039 

5.18 

.64481 

3-51 

•25562 

4.07 

i  .  40364 

4.63 

1-53256 

5.19 

•  64673 

3.52 

.25846 

4.08 

1.40610 

4-64 

I-5347I 

5.20 

.64866 

3-53 

.26130 

4-09 

i  .  40854 

4.65 

1-53687 

5.21 

•65058 

3-54 

.26412 

4.10 

1.41099 

4.66 

1-53902 

5.22 

•65250 

3-55 

.26695 

4.11 

1.41342 

4.67 

1.54116 

5.23 

.65441 

3.56 

.26976 

4.12 

1-41585 

4.68 

i  •  54330 

5.24 

-65632 

3-57 

.27257 

4.13 

1.41828 

4.69 

1-54543 

5-25 

•65823 

3.58 

•27536 

4.14 

1.42070 

4.70 

L54756 

5.26 

.66013 

3-59 

.27815 

4.15 

1.42311 

4.71 

1.54969 

i  5.27 

.  66203 

3.6o 

.28093 

4.16 

1-42552 

4.72 

i-55i8i 

5.28 

-66393 

3.6i 

.28371 

4.17 

1.42792 

4-73 

1-55393 

5.29 

.  66582 

3.62 

.28647 

4.18 

1-43031 

4.74 

1.55604 

5.30 

.66771 

3.63 
3.64 

•28923 
.29198 

4.19 
4.20 

1.43270 
i  .  43508 

4.76 

i-558i4 
1.56025 

5.3i 
5.32 

-66959 
•67147 

3.65 

•29473 

4.21 

1.43746 

4.77 

1-56235 

5-33 

•67335 

3-66 

.29746 

4.22 

i  .  43984 

4.78 

1.56444 

5-34 

•67523 

3.67 

.30019 

4.23 

1.44220 

4.79 

1-56653 

5-35 

.67710 

3-68 

.30291 

4.24 

1.44456 

4.80 

i  .  56862 

5.36 

.  67896 

3.69 

•  30563 

4.25 

i  .  44692 

4.81 

1.57070 

5-37 

.68083 

3.70 

•30833 

4.26 

1.44927 

4.82 

I-57277 

5.38 

.68269 

3.7i 

•31103 

4.27 

1.45161 

4.83 

I.57485 

5-39 

.68455 

3.72 

•3!372 

4.28 

1-45395 

4.84 

1.57691 

5-40 

.68640 

3-73 

.31641 

4-29 

1.45629 

4-85 

1.57898 

5.4i 

.68825 

3-74 

.31909 

4.30 

1.45861 

4.86 

i  .58104 

5.42 

.69010 

3-75 

.32176 

4.31 

1.46094 

4.87 

1.58309 

5-43 

.69194 

3.76 

.32442 

4.32 

1.46326 

4.88 

1-58515 

5-44 

.69378 

3-77 

.32707 

4-33 

1-46557 

4-89 

1.58719 

5-45 

.69562 

3.78 

•32972 

4-34 

1.46787 

4.90 

1.58924 

5.46 

-69745 

3.79 

•33237 

4.35 

i  .47018 

4.91 

1.59127 

5.47 

.69928 

3.8o 

•335°o 

4-36 

1.47247 

4.92 

i  -5933  i 

5.48 

.70111 

TABLES 


149 


TABLE  XIII.    Continued. — HYPERBOLIC  LOGARITHMS 


N. 

Loga- 
rithm. 

N. 

Loga- 
rithm. 

X. 

Loga- 
rithm. 

N. 

Loga- 
rithm. 

5-49 

.70293 

6.05 

1.80006 

6.61 

.88858 

7.17 

1.96991 

5-50 

•  70475 

6.06 

1.80171 

6.62 

.89010 

7.18 

1.97130 

5.51 

•70656 

6.07 

1.80336 

6.63 

.89160 

7.19 

1.97269 

5.52 

.70838 

6.08 

1.80500 

6.64 

.89311 

7.20 

1.97408 

5-53 

.71019 

6.09 

1.80665 

6.65 

.89462 

7.21 

1  -97547 

5-54 

.71199 

6.10 

1.80829 

6.66 

.89612 

7.22 

1.97685 

5-55 

.71380 

6.  ii 

1.80993 

6.67 

.89762 

7-23 

1.97824 

5.56 

.71560 

6.12 

1.81156 

6.68 

.89912 

7.24 

1.97962 

5.57 

.71740 

6.13 

1.81319 

6.69 

.90061 

7-25 

1.98100 

5-58 

.71919 

6.14 

1.81482 

6.70 

.  902  i  i 

7.26 

1.98238 

5-59 

.  72098 

6.15 

1.81645 

6.71 

.90360 

7.27 

1.98376 

5.6o 

.72277 

6.16 

i.  81808 

6.72 

•90509 

7.28 

•72455 

6.17 

1.81970 

6-73 

.90658 

7.29 

1.98650 

5^2 

•72633 

6.18 

1.82132 

6.74 

.90806 

7.30 

1.98787 

5.63 

.72811 

6.19 

1.82294 

6-75 

.90954 

7-31 

1.98924 

5-64 

.  72988 

6.20 

1-82455 

6.76 

.91102 

7-32 

1.99061 

5.65 

.73166 

6.21 

1.82616 

6.77 

.91250 

7-33 

1.99198 

5-66 

•  73342 

6.22 

1.82777 

6.78 

.91398 

7-34 

1-99334 

5.67 

•73519 

6.23 

1.82937 

6.79 

•91545 

7-35 

1.99470 

5.68 

•  73695 

6.24 

i  .  83098 

6.80 

.91692 

7.36 

1.99606 

5.69 

•73871 

6.25 

1.83258 

6.81 

.91839 

7-37 

1.99742 

5-70 

.  74047 

6.26 

1.83418 

6.82 

.91986 

7.38 

1.99877 

5-71 

.74222 

6.27 

I-83578 

6.83 

.92132 

7-39 

2.00013 

5.72 

•74397 

6.28 

1  -83737 

6.84 

.92279 

7.40 

2.00148 

5-73 

•74572 

6.29 

1.83896 

6.85 

.92425 

7.41 

2.00283 

5-74 

.74746 

6.30 

1.84055 

6.86 

.92571 

7.42 

2.00418 

5-75 

6.3I 

1.84214 

6.87 

.92716 

7-43 

2.00553 

5.76 

•  75°94 

6.32 

1.84372 

6.88 

.92862 

7-44 

2.00687 

5-77 

•75267 

6-33 

i  .  84530 

6.89 

.93007 

7-45 

2.00821 

5.78 

•  75440 

6.34 

1.84688 

6.90 

•93152 

7.46 

2.00956 

5-79 

•75613 

6-35 

1.84845 

6.91 

•93297 

7.47 

2.01089 

5.8o 

•75786 

6.36 

1.85003 

6.92 

•93442 

7.48 

2.01223 

5-8i 

•7595s 

6.37 

1.85160 

6-93 

•93586 

7-49 

2-01357 

5-82 

.76130 

6.38 

I.853I7 

6.94 

•93730 

7-50 

2.01490 

5-83 

•  76302 

6-39 

1*85473 

6.95 

•93874 

7.51 

2.01624 

5.84 

•  76473 

6.40 

1.85630 

6.96 

.94018 

7.52 

2-01757 

5.85 

.  76644 

6.41 

1.85786 

6.97 

.94162 

7-53 

2.01890 

5-86 

•76815 

6.42 

1.85942 

6.98 

•94305 

7-54 

2.02022 

5.87 

.76985 

6.43 

1.86097 

6.99 

•94448 

7-55 

2.02!55 

5.88 

•77156 

6-44 

1.86253 

7.00 

•94591 

7-56 

2.02287 

5-89 

•77326 

6-45 

1.86408 

7.01 

•94734 

7-57 

2.02419 

5-90 

.  77495 

6.46 

1.86563 

7.02 

.94876 

7.58 

2.02551 

5-91 

•77665 

6.47 

1.86718 

7-03 

.95019 

7-59 

2.02683 

5-92 

•77834 

6.48 

1.86872 

7.04 

.95161 

7.60 

2.02815 

5-93 

.  78002 

6.49 

1.87026 

7-05 

•95303 

7.61 

2.02946 

5-94 

.78171 

6.50 

1.87180 

7.06 

•  95444 

7.62 

2.03078 

5-95 

•  78339 

6.51 

I.87334 

7.07 

.95586 

7-63 

2.03209 

5.96 

•  78507 

6.52 

1.87487 

7.08 

•95727 

7.64 

2.03340 

5-97 

•786/5 

6-53 

1.87641 

7.09 

.95869 

7.65 

2.03471 

5.98 

.  78842 

6-54 

1.87794 

7.10 

.96009 

7.66 

2.03601 

5-99 

.  79009 

6-55 

1.87947 

7.11 

.96150 

7.67 

2.03732 

6.00 

.79176 

6.56 

i  .  88099 

7.12 

96291 

7.68 

2.03862 

6.01 

•  79342 

6.57 

1.88251 

7.13 

96431 

7.69 

2.03992 

6.02 

•  79509 

6.58 

1.88403 

7.14 

96571 

7.70 

2.O4I22 

6.03 

•  79675 

6-59 

1-88555 

7-15 

96711 

7.71 

2.04252 

6.04 

.  70)840 

6.60 

i  88707 

7.16 

968  si 

7.72 

2.0438r 

150 


COMPRESSED  AIR 


TABLE  XIII.  Continued. — HYPERBOLIC  LOGARITHMS 


N. 

Loga- 
rithm. 

N. 

Loga- 
rithm. 

N. 

Loga- 
rithm. 

N. 

Loga- 
rithm. 

7-73 

2  .  045  1  1 

8.30 

2.  11626 

8.87 

2  .  18267 

9.44 

2.24496 

7-74 

2  .  04640 

8.31 

2.11746 

8.88 

2.18380 

9-45 

2  .  24601 

7-75 

2.04769 

8.32 

2.II866 

8.89 

2.18493 

9.46 

2.24707 

7.76 

2.04898 

8.33 

2.11986 

8.90 

2.18605 

9-47 

2.24813 

7-77 

2.05027 

8.34 

2.  I2I06 

8.91 

2.18717 

9.48 

2.24918 

7.78 

2.05156 

8-35 

2.12226 

8.92 

2.18830 

9-49 

2.25024 

7-79 

2.05284 

8.36 

2.12346 

8.93 

2  .  18942 

9.50 

2.25129 

7.80 

2.05412 

8.37 

2.  12465 

8.94 

2.19054 

9-51 

2-25234 

7.81 

2.05540 

8.38 

2.12585 

8-95 

2.19165 

9.52 

2.25339 

7.82 

2.05668 

8.39 

2.  12704 

8.96 

2.19277 

9-53 

2.25444 

7-83 

2.05796 

8.40 

2.12823 

8.97 

2.19389 

9-54 

2.25549 

7.84 

2.05924 

8.41 

2.  12942 

8.98 

2.19500 

9-55 

2-25654 

7.85 

2.06051 

8.42 

2.  13061 

8.99 

2  .  19611 

9-56 

2.25759 

7.86 

2.06179 

8.43 

2.13180 

9.00 

2.19722 

9-57 

2.25863 

7.87 

2.06306 

8.44 

2.13298 

9.01 

2.19834 

9.58 

2  .25968 

7.88 

2.06433 

8.45 

2.I34I7 

9.02 

2.19944 

9-59 

2  .26072 

7.89 

2.06560 

8.46 

2-13535 

9-03 

2.20055 

9.60 

2.26176 

7.90 

2.06686 

8.47 

2.13653 

9.04 

2.2OI66    1 

9.61 

2.26280 

7.91 

2.06813 

8.48 

2.I377I 

9-05 

2.2O276 

9.62 

2.26384 

7.92 

2.06939 

8.49 

2.13889 

9.06 

2.20387 

9-63 

2.26488 

7-93 

2  .  07065 

8.50 

2.  14007 

9.07 

2.20497 

9.64 

2  .26592 

7-94 

2.07191 

8.51 

2.I4I24 

9.08 

2  .20607 

9.65 

2  .  26696 

7-95 

2.07317 

8.52 

2.14242 

9.09 

2.20717 

9.66 

2.  26799 

7.96 

2.07443 

8-53 

2-14359 

9.10 

2  .20827 

9.67 

2.26903 

7-97 

2.07568 

8.54 

2.14476 

9.11 

2.20937 

9.68 

2  .27006 

7.98 

2.07694 

8.55 

2.14593 

9.12 

2.21047 

9.69 

2.27109 

7-99 

2.07819 

8.56 

2.  I47IO 

9.13 

2.2II57 

9.70 

2     27213 

8.00 

2.07944 

8.57 

2.14827 

9.14 

2.21266 

9.71 

2.27316 

8.01 

2.08069 

8.58 

2  .  14943 

9-15 

2.21375 

9.72 

2.27419 

8.02 

2.08194 

8.59 

2.  15060 

9.16 

2.21485 

9-73 

2.27521 

8.03 

2.0831,8 

8.60 

2.15176 

9.17 

2.21594 

9-74 

2.27624 

8.04 

2.08443 

8.61 

2.15292 

9.18 

2.21703 

9-75 

2.27727 

8.05 

2.08567 

8.62 

2.15409 

9.19 

2.  2l8l2 

9.76 

2.27829 

8.06 

2.08691 

8.63 

2.15524 

9.20 

2.2I92O 

9-77 

2.27932 

8.07 

2.08815 

8.64 

2.15640 

9.21 

2.  22O29 

9.78 

2.28034 

8.08 

2.08939 

8.65 

2-I5756 

9.22 

2.22138 

9-79 

2.28136 

8.09 

2.09063 

8.66 

2.15871 

9-23 

2  .22246 

9.80 

2.28238 

8.10 

2.09186 

8.67 

2.15987 

9.24 

2.22351 

9.81 

2.28340 

8.ii 

2.09310 

8.68 

2.  l6l02 

9-25 

2.22462 

9.82 

2  .28442 

8.12 

2  .  09433 

8.69 

2.I62I7 

9.26 

2.2257O 

9-83 

2.28544 

8.13 

2.09556 

8.70 

2.16332 

9.27 

2.22678 

9.84 

2.28646 

8.14 

2.09679 

8.71 

2.16447 

9.28 

2.22786 

9-85 

2.28747 

8.15 

2  .  09802 

8.72 

2.  16562 

9.29 

2.22894 

9.86 

2.28849 

8.16 

2.09924 

8-73 

2.16677 

9-30 

2.23001 

9.87 

2.28950 

8.17 

2  .  10047 

8.74 

2.16791 

9-31 

2.23109 

9.88 

2.29051 

8.18 

2.10169 

8-75 

2.16905 

9-32 

2.23216 

9.89 

2.29152 

8.19 

2.I029I 

8.76 

2.17020 

9-33 

2.23323 

9.90 

2.29253 

8.20 

2.I04I3 

8.77 

2.I7I34 

9-34 

2.23431 

9.91 

2.29354 

8.21 

2.10535 

8.78 

2  .  17248 

9-35 

2.23538 

9-92 

2.29455 

8.22 

2.10657 

8.79 

2.17361 

9-30 

2-23645 

9-93 

2.29556 

8.23 

2.10779 

8.80 

2.17475 

9-37 

2.23751 

9-94 

2.29657 

8.24 

2  .  IO9OO 

8.81 

2.17589 

9.38 

2.23858 

9-95 

2.29757 

8.25 

2.IIO2I 

8.82 

2.  I77O2 

9-39 

2.23965 

9.96 

2.29858 

8.26 

2.III42 

8.83 

2.17816 

9.40 

2.24O7I 

9-97 

2.29958 

8.27 

2.11263 

8.84 

2.17929 

9.41 

2.24177 

9.98 

2  .30058 

8.28 

2.11384 

8.85 

2.18042 

9.42 

2.24284 

9.99 

2.30158 

8.29 

2.nso<; 

8.86 

2.l8l55 

9^43_ 

2  .  24390 

TABLES 
TABLE  XIV.— LOGARITHMS  OF  NUMBERS 


151 


No. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

Pp.  Pts. 

100 

OO  000 

°43 

087 

130 

173 

217 

260 

303 

346 

389 

IOI 

432 

475 

518 

604 

647 

689 

732 

775 

*817 

102 

860 

9°3 

945 

988 

030 

*072 

"5 

199 

44 

43 

42 

103 

01  284 

326 

368 

410 

452 

494 

536 

578 

620 

662 

i 

2 

s's 

8*6 

K 

104 

7°3 

745 

787 

828 

870 

912 

953 

995 

*©36 

^078 

3 

13-2 

12.9 

12.6 

105 

02  119 

1  60 

202 

243 

284 

325 

366 

407 

449 

490 

4 

17.6 

17.2 

16.8 

1  06 

531 

572 

612 

653 

694 

735 

776 

816 

857 

898 

I 

22.  O 
26.4 

21.5 
25-8 

21.  0 
25.2 

107 

938 

979 

*oi9 

*o6c 

*IOO 

*i8i 

*222 

*262 

*302 

I 

30.8 

30.1 

29.4 

1  08 

03  342 

383 

423 

463 

5°3 

543 

583 

623 

663 

7°3 

8 
9 

35-2 
39-6 

34-4 
38.7 

33-6 
37.8 

IO9 

743 

782 

822 

862 

902 

941 

981 

*O2I 

*o6o 

*IOO 

110 

04  139 

179 

218 

258 

297 

336 

376 

415 

454 

493 

III 

532 

571 

610 

650 

689 

727 

766 

805 

844 

883 

112 

922 

961 

999 

*o38 

*o77 

*«5 

*I54 

*I92 

*23I 

*269 

j 

41 

40 

39 

113 

05  308 

346 

385 

423 

461 

500 

538 

576 

614 

652 

2 

4*  * 

8.2 

4*  o 
8.0 

3  •  9 

7.8 

114 

690 

729 

767 

805 

843 

881 

918 

956 

994 

*032 

3 

12.3 

12.0 

ii.  7 

1*5 

06  070 

108 

183 

221 

258 

296 

333 

37! 

408 

4 
5 

16.4 
20.5 

16.0 

20.0 

15-6 
19-5 

116 

446 

483 

521 

558 

595 

*633 

*67° 

707 

744 

78l 

6 

24.6 

24-0 

23-4 

117 

819 

856 

893 

93° 

967 

*0?8 

*"5 

*I5I 

g7 

28.7 
32.8 

28.0 
32.0 

27-3 
31.2 

118 

07  188 

225 

262 

298 

335 

372 

408 

445 

482 

518 

9 

36.9 

36.0 

35-  1 

119 

555 

591 

628 

664 

700 

737 

773 

809 

846 

882 

120 

9i§ 

954 

990 

*027 

*o63 

*°99 

*i35 

*i7i 

*207 

*243 

121 

08  279 

314 

35° 

386 

422 

458 

493 

529 

565 

600 

16 

122 

636 

672 

707 

*743 

778 

814 

849 

884 

920 

*955 

i 

3-8 

3-7 

3  £ 

3-6 

123 

991 

*026 

*o6i 

*I32 

*i67 

*202 

*237 

*272 

2 

7-6 

7-4 

7-2 

I24 

09  342 

377 

412 

447 

482 

552 

587 

621 

656 

3 

4 

11.4 
15.2 

ii.  i 
14.8 

10.8 
14.4 

125 

691 

726 

760 

795 

830 

864 

899 

934 

968 

*oo3 

5 

19.0 

18.5 

18.0 

126 

10  037 

072 

106 

140 

175 

209 

243 

278 

312 

346 

6 
7 

22.8 
26.6 

22.  2 
2S.9 

21.6 

25.2 

127 

380 

4T5 

449 

483 

517 

551 

585 

619 

653 

*027 

8 

30.4 

29.6 

28.8 

128 

72! 

755 

789 

823 

857 

890 

924 

958 

992 

9 

34-2 

33-3 

32.4 

129 

II  059 

°93 

126 

1  60 

193 

227 

26l 

294 

327 

361 

130 

394 

428 

461 

494 

528 

561 

594 

628 

66  1 

694 

727 

760 

793 

826 

860 

893 

926 

959 

992 

*024 

35 

34 

33 

132 

12  057 

090 

123 

156 

l89 

222 

254 

287 

320 

352 

i 

3-5 

3-4 

3-3 

133 

385 

418 

45° 

483 

5l6 

548 

613 

646 

678 

2 

3 

7.0 
10.5 

6.8 

10.2 

6.6 
9-9 

134 

710 

743 

775 

808 

840 

872 

905 

937 

969 

*OOI 

4 

14.0 

13-6 

13-2 

135 

13  °33 

066 

098 

130 

162 

194 

226 

258 

290 

322 

i 

I7-S 

21.  0 

17.0 
20.4 

16.5 
19.8 

136 

354 

386 

418 

45° 

481 

513 

545 

577 

609 

640 

7 

24-5 

23.8 

23.1 

137 
138 

672 

988 

704 

735 
*o5i 

767 

*082 

*799 

830 

*I45 

862 
*I76 

893 

*208 

925 
*239 

956 

*270 

8 
9 

28.0 
31-5 

27.2 
30.6 

26.4 
29-7 

139 

14  301 

333 

364 

395 

426 

457 

489 

520 

551 

582 

140 

613 

644 

675 

706 

737 

768 

799 

829 

860 

89I 

141 

922 

953 

983 

*oi4 

*<H5 

=•=076 

*io6 

*i37 

*i68 

*i98 

32 

31 

30 

142 

15  229 

259 

290 

320 

35  I 

381 

412 

442 

473 

5°3 

i 

2 

3-2 

6.4 

1:1 

3-0 
6.0 

143 

53Z 

564 

594 

625 

655 

685 

7J5 

746 

776 

806 

3 

9.6 

9-3 

9.0 

144 

836 

866 

89 

92 

957 

987 

*oi7 

*077 

*I07 

4 
5 

12.8 

i6.c 

12.4 
15.5 

12.0 

15.0 

145 

16  137 

16 

19 

22 

25^ 

286 

316 

346 

376 

406 

6 

19.2 

18.6 

18.0 

I46 

435 

465 

495 

32i 

554 

584 

613 

643 

673 

702 

"; 

22.4 

21.7 

_  .   0 

21.0 

147 

732 

76 

79 

820 

879 

909 

93s 

967 

997 

9 

25.  c 

28.8 

24*  * 

27.5 

24.  o 
27.0 

I48 

17  026 

°5^ 

085 

114 

143 

173 

202 

23 

260 

289 

I49 

3*9 

348 

37 

40^ 

435 

464 

493 

522 

551 

580 

152  COMPRESSED  AIR 

TABLE  XIV.  Continued. — LOGARITHMS  OF  NUMBERS 


No. 

o 

I 

2 

3 

4 

5 

6 

7 

8 

9 

Pp.  PtB. 

ISO 

17  609 

638 

667 

696 

725 

754 

*o8o 

8n 

840 

869 

898 

926 

955 

984 

*0!3 

*099 

*I27 

*i56 

152 

18  184 

213 

241 

270 

298 

327 

355 

384 

412 

441 

29 

28 

153 

469 

498 

526 

554 

583 

611 

639 

667 

696 

724 

I 
2 

2.9 

c  g 

2.8 

154 

752 

780 

808 

837 

865 

893 

92! 

949 

977 

3 

5  •  ° 
8-7 

i.1 

155 

19  033 

061 

089 

117 

145 

173 

201 

229 

257 

285 

4 

ii.  6 

II.  2 

156 

312 

340 

368 

396 

424 

479 

5°7 

535 

562 

5 
6 

14-5 
17-4 

14.0 

16.8 

157 

59° 

618 

645 

673 

700 

728 

He756 

*783 

8n 

838 

I 

20.3 

19.6 

158 

866 

893 

921 

948 

976 

*oo3 

*o85 

*II2 

8 
9 

23-2 
26.1 

22.4 

2<C.2 

159 

20  140 

167 

194 

222 

249 

276 

303 

330 

358 

385 

1  60 

412 

439 

466 

493 

520 

548 

575 

602 

629 

656 

161 

683 

710 

737 

*?63 

790 

817 

844 

871 

898 

925 

162 

952 

978 

*o85 

*II2 

*i39 

*i65 

*I92 

27 

26 

2  6 

163 

21  219 

245 

272 

299 

325 

352 

378 

4°5 

431 

458 

2 

2.  7 
5-4 

5-2 

164 

484 

5" 

537 

564 

59° 

617 

643 

669 

696 

722 

3 

8.1 

7-8 

165 

748 

775 

801 

827 

854 

880 

906 

932 

958 

985 

4 
5 

10.8 
13-5 

10.4 
13.0 

166 

22  Oil 

°37 

063 

089 

141 

l67 

194 

220 

246 

6 

16.2 

15.6 

167 

272 

298 

324 

35° 

376 

401 

427 

453 

479 

5°5 

I 

18.9 

21.6 

18.2 

20.8 

168 

531 

557 

583 

608 

634 

660 

686 

7!2 

737 

*763 

9 

24.3 

23.4 

169 

789 

814 

840 

866 

89I 

917 

943 

968 

994 

170 

23  045 

070 

096 

121 

147 

172 

198 

223 

249 

274 

171 

300 

325 

350 

376 

401 

426 

452 

477 

502 

528 

172 

553 

603 

629 

654 

679 

704 

729 

754 

779 

i  2.5 

173 

805 

830 

855 

880 

9°5 

93° 

955 

980 

2  5.0 

174 

24  055 

080 

I30 

155 

1  86 

204 

229 

254 

279 

3  7-5 
4  10.  o 

175 

3°4 

329 

353 

378 

403 

428 

452 

477 

502 

527 

512.5 

176 

576 

601 

625 

650 

674 

699 

724 

748 

773 

o  15.0 
7  i7-5 

177 

797 

822 

846 

87I 

895 

920 

944 

969 

993 

*oi8 

o 

O  2O.O 

178 

25  042 

066 

091 

115 

139 

164 

188 

212 

237 

261 

9  22.5 

179 

285 

310 

334 

358 

382 

406 

431 

455 

479 

5°3 

1  80 

527 

551 

575 

600 

624 

648 

672 

696 

720 

744 

181 

768 

792 

816 

840 

864 

888 

912 

935 

959 

983 

24 

23 

182 

26  007 

031 

055 

079 

1  02 

126 

15° 

174 

198 

221 

i 

2.4 

2.3 

183 

245 

269 

293 

340 

364 

387 

411 

435 

458 

2 
3 

4.8 
7  2 

4-6 
6.9 

184 

482 

5°5 

529 

553 

576 

600 

623 

647 

670 

694 

4 

9.6 

9-2 

185 

717 

74i 

764 

788 

811 

834 

858 

881 

90S 

928 

9 

12.0 

ii.  5 

186 

975 

998 

*02I 

*Q45 

*o68 

*H4 

*i38 

*i6i 

o 

7 

14.4 

16.8 

13.  8 
16.  i 

187 

27  184 

207 

231 

254 

277 

300 

323 

346 

37° 

393 

8 

19.2 

18.4 

1  88 

416 

439 

462 

485 

508 

554 

577 

600 

623 

9 

21.6 

20.7 

189 

646 

669 

692 

715 

738 

761 

784 

*8°7 

830 

852 

190 

875 

898 

921 

944 

967 

989 

*OI2 

1=058 

*o8i 

191 

28  103 

126 

149 

171 

194 

217 

240 

262 

285 

307 

22 

21 

192 
193 

33° 
556 

353 
578 

375 
601 

398 
623 

421 
646 

443 
668 

466 
691 

488 

5" 
735 

533 

758 

i 

2 

3 

2.2 

4-4 
6.6 

2.  I 
4-2 

6-3 

194 

780 

803 

825 

847 

870 

892 

914 

937 

959 

981 

4 

8.8 

8.4 

195 

29  003 

026 

048 

070 

092 

"5 

137 

159 

181 

203 

i 

II.  O 

13-2 

10.5 

12.6 

196 

226 

248 

270 

292 

336 

358 

380 

403 

425 

j 

15-4 

14.7 

197 

447 

469 

491 

513 

535 

557 

579 

601 

623 

645 

o 
9 

17.6 
19.8 

16.  8 
18.9 

198 

667 

688 

710 

732 

754 

776 

798 

820 

842 

863 

199 

885 

907 

929 

973 

994 

*oi6 

1=038 

*o6o 

*o8i 

3,0 

TABLES 
TABLE  XIV.  Continued. — LOGARITHMS  OF  NUMBERS 


153 


No. 

O 

i 

2 

3 

4 

5 

6 

7 

8 

9 

Pp.  Pts. 

20O 

30  103 

i25 

146 

168 

190 

211 

233 

255 

276 

298 

201 

320 

34i 

363 

384 

406 

428 

449 

471 

492 

5M 

202 

535 

557 

578 

600 

621 

643 

664 

685 

707 

728 

22   21 

203 

75° 

771 

792 

814 

835 

856 

878 

899 

920 

942 

I   2.2   2.1 

2    A  A   A  2 

204 

963 

984 

*oo6 

*027 

*048 

*o6g 

*c»9i 

*II2 

*i33 

*I54 

4-4  4-  z 

3  6.6  6.3 

205 

3i  175 

197 

218 

239 

260 

281 

302 

323 

345 

366 

4  8.8  8.4 

206 

387 

408 

429 

45° 

47i 

492 

513 

534 

555 

576 

5  ii.  o  10.5 
5  13.2  12.6 

207 

597 

618 

639 

660 

681 

702 

723 

744 

765 

785 

7  15.4  14.7 

208 

806 

827 

848 

869 

890 

911 

93i 

952 

973 

994 

3  17.6  16.8 
9  19.8  18.9 

20Q 

32  015 

°35 

056 

077 

098 

118 

139 

1  60 

181 

201 

210 

222 

243 

263 

284 

3°5 

325 

346 

366 

387 

408 

211 

428 

449 

469 

49° 

51° 

53i 

55'2 

572 

593 

613 

212 

634 

654 

675 

695 

7*5 

736 

.756 

777 

797 

818 

20 

213 

838 

858 

879 

899 

919 

940 

960 

980 

*OOI 

*02I 

2 

4.0 

214 

33  °4i 

062 

082 

102 

122 

143 

163 

183 

203 

224 

3 

6.0 

215 

244 

264 

284 

3°4 

325 

345 

365 

385 

405 

425 

4 
5 

8.  o 

10.  0 

216 

445 

465 

486 

506 

526 

546 

566 

586 

606 

626 

6 

12.0 

217 

646 

666 

686 

706 

726 

746 

766 

786 

806 

826 

I 

14.0 

16.0 

218 

846 

866 

885 

9°5 

925 

945 

965 

985 

*oo5 

*025 

9 

18.0 

2IQ 

34  044 

064 

084 

104 

124 

143 

163 

183 

203 

223 

220 

242 

262 

282 

301 

321 

34i 

361 

380 

400 

420 

221 

439 

459 

479 

498 

518 

537 

557 

577 

596 

616 

222 

635 

655 

674 

694 

713 

733 

753 

772 

792 

811 

I 

19 

1.9 

223 

830 

850 

869 

889 

908 

928 

947 

967 

986 

*oo5 

2 

3.8 

224 

35  025 

044 

064 

083 

102 

122 

141 

160 

180 

199 

3 

4 

H 

225 

218 

238 

257 

276 

295 

3i5 

334 

353 

372 

392 

5 

9-5 

226 

411 

43° 

449 

468 

488 

5°7 

526 

545 

564 

585 

6 

11.4 

227 

603 

622 

641 

660 

679 

698 

717 

736 

755 

774 

I 

*3-  3 
IS-2 

228 

793 

813 

832 

851 

870 

889 

908 

927 

946 

965 

9 

17.1 

229 

984 

*oo3 

*O2I 

*040 

*°59 

*078 

*097 

*n6 

*i35 

*i54 

230 

36  173 

192 

211 

229 

248 

267 

286 

3°5 

324 

342 

231 

361 

380 

399 

418 

436 

455 

474 

493 

5ii 

53° 

18 

232 

549 

568 

586 

605 

624 

642 

661 

680 

698 

717 

i 

1.8 

233 

736 

754 

773 

791 

810 

829 

847 

866 

884 

903 

2 
3 

3-6 
5  •  4 

234 

922 

940 

959 

977 

996 

*oi4 

*033 

*°5i 

*o7o 

*o88 

4 

7.2 

235 

37  *°7 

I25 

144 

162 

181 

199 

218 

236 

254 

273 

| 

9.0 

10  8 

236 

291 

310 

328 

346 

365 

383 

401 

420 

438 

457 

7 

12^6 

237 

475 

493 

5n 

53° 

548 

566 

585 

603 

621 

639 

8 

14.4 

238 

658 

676 

694 

712 

73i 

749 

767 

785 

803 

822 

9 

16.  a 

239 

840 

858 

876 

894 

912 

93i 

949 

967 

985 

*oo3 

240 

38  02  1 

039 

057 

°75 

093 

112 

130 

148 

1  66 

184 

241 

202 

220 

238 

256 

274 

292 

310 

328 

346 

364 

17 

242 

382 

399 

4i7 

435 

453 

471 

489 

5°7 

525 

543 

I 

2 

i-7 

3   A 

243 

561 

578 

596 

614 

632 

650 

668 

686 

7°3 

721 

3 

•  *r 

S-i 

244 

739 

757 

775 

792 

810 

828 

846 

863 

881 

899 

4 

6.8 

245 

917 

934 

952 

970 

987 

*oo5 

*023 

*o4i 

*058 

*076 

! 

8.5 

10.2 

246 

39  °94 

in 

129 

146 

164 

182 

199 

217 

235 

252 

' 

ii.  9 

247 

270 

287 

3°5 

322 

340 

358 

375 

393 

410 

428 

o 
9 

13-6 
15.3 

248 

445 

463 

480 

498 

5i5 

533 

55° 

568 

585 

602 

249 

620 

637 

655 

672 

690 

707 

724 

742 

759 

777 

154  COMPRESSED  AIR 

TABLE  XIV.  Continued. — LOGARITHMS  OF  NUMBERS 


No. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

Pp.  PtB. 

250 

39  794 

8n 

829 

846 

863 

881 

898 

9i5 

933 

95° 

251 

967 

985 

*002 

*oi9 

*o37 

*°54 

*o7i 

*o88 

*io6 

*I23 

252 

40  140 

157 

175 

192 

209 

226 

243 

261 

278 

295 

18 

253 

312 

329 

346 

364 

381 

398 

4i5 

432 

449 

466 

I 

1.8 

254 

483 

500 

5l8 

535 

552 

569 

586 

603 

620 

637 

2 

3 

3-6 

5-  4 

255 

654 

671 

688 

7°5 

722 

739 

756 

773 

790 

807 

4 

7-2 

256 

824 

841 

858 

875 

892 

909 

926 

943 

960 

976 

I 

9.0 
10.8 

257 

993 

*OIO 

*027 

*044 

*o6i 

*078 

*095 

*in 

*I28 

*I45 

7 

12.6 

258 

41  162 

179 

196 

212 

229 

246 

263 

280 

296 

3i3 

8 
g 

14.4 

TfS  1 

259 

330 

347 

363 

380 

397 

414 

430 

447 

464 

481 

260 

497 

5J4 

531 

547 

564 

58i 

597 

6x4 

631 

647 

261 

664 

681 

697 

714 

73i 

747 

764 

780 

797 

814 

262 

830 

847 

863 

880 

896 

9i3 

929 

946 

963 

979 

17 

263 

996 

*OI2 

*029 

*°45 

*062 

$078 

*095 

*in 

*I27 

*I44 

I 

2 

1  •  7 
3  •  4 

264 

42  1  60 

177 

193 

210 

226 

243 

259 

275 

292 

308 

3 

5-i 

265 

325 

341 

357 

374 

39° 

406 

423 

439 

455 

472 

4 

^ 

6.8 

8.5 

266 

488 

5°4 

521 

537 

553 

57° 

586 

602 

619 

635 

6 

10.2 

267 

651 

667 

684 

700 

716 

732 

749 

765 

78i 

797 

7 

8 

II.  9 

j.»  6 

268 

813 

830 

846 

862 

878 

894 

911 

927 

943 

959 

9 

Ao  -  u 
IS-3 

269 

975 

991 

*oo8 

*O24 

*040 

*056 

*072 

*o88 

*io4 

*I2O 

270 

43  i36 

IS2 

169 

185 

201 

217 

233 

249 

265 

28l 

271 

297 

3i3 

329 

345 

361 

377 

393 

409 

425 

441 

-A 

272 

457 

473 

489 

5°5 

521 

537 

553 

569 

584 

600 

i 

IO 

1.6 

273 

616 

632 

648 

664 

680 

696 

712 

727 

743 

759 

2 

3-2 

274 

775 

791 

807 

823 

838 

854 

870 

886 

902 

917 

3 

A 

4.8 
6.4 

275 

933 

949 

965 

981 

996 

*OI2 

*028 

*o44 

*059 

*o75 

5 

8.0 

276 

44  091 

107 

122 

138 

154 

170 

185 

2OI 

217 

232 

6 

7 

9.6 

II  2 

277 

248 

264 

279 

295 

311 

326 

342 

358 

373 

389 

8 

12^8 

278 

404 

420 

436 

45i 

467 

483 

498 

514 

529 

545 

9 

14.4 

279 

560 

576 

592 

607 

623 

638 

654 

669 

685 

700 

280 

716 

73i 

747 

762 

778 

793 

809 

824 

840 

855 

281 

871 

886 

902 

917 

932 

948 

963 

979 

994 

*OIO 

15 

282 

45  025 

040 

056 

071 

086 

102 

117 

i33 

148 

163 

i 

i-5 

283 

179 

194 

209 

225 

240 

255 

271 

286 

301 

3J7 

2 

3-o 

A,  CJ 

284 

332 

347 

362 

378 

393 

408 

423 

439 

454 

469 

4 

6.0 

285 

484 

500 

5*5 

530 

545 

56l 

576 

59i 

606 

621 

i 

7-5 

286 

637 

652 

667 

682 

697 

712 

728 

743 

758 

773 

7 

9.0 
10.5 

287 

788 

803 

818 

834 

849 

864 

879 

894 

909 

924 

8 

12.0 

288 

939 

954 

969 

984 

*000 

*oi5 

*030 

*Q45 

*o6o 

*Q75 

9 

13-5 

289 

46  090 

io5 

120 

i35 

150 

165 

1  80 

195 

210 

225 

290 

240 

255 

270 

285 

300 

3i5 

33° 

345 

359 

374 

291 

389 

404 

419 

434 

449 

464 

479 

494 

509 

523 

14 

292 

538 

553 

S68 

583 

598 

613 

627 

642 

657 

672 

i 

2 

i-4 

2.8 

293 

687 

702 

7l6 

73i 

746 

761 

776 

790 

805 

820 

3 

4-2 

294 

835 

850 

864 

879 

894 

909 

923 

938 

953 

967 

4 

5-6 

295 

982 

997 

*OI2 

*026 

*04i 

*©56 

+070 

*o85 

*IOO 

*n4 

6 

7.0 
8.4 

296 

47  I29 

144 

159 

173' 

188 

202 

217 

232 

246 

261 

z 

9.8 

297 

276 

290 

3°5 

3i9 

334 

349 

363 

378 

392 

407 

o 
9 

II  .  2 
12.6 

298 

422 

436 

45  * 

465 

480 

494 

5°9 

524 

538 

553 

299 

567 

582 

596 

611 

625 

640 

654 

669 

683 

698 

TABLES 
TABLE  XIV.  Continued.— LOGARITHMS  OF  NUMBERS 


155 


No. 

o 

i 

2 

3 

4 

5 

6 

7 

8 

9 

Pp.  Pt8. 

300 

47  712 

727 

741 

756 

770 

784 

799 

813 

828 

842 

301 

857 

871 

885 

900 

914 

929 

943 

958 

972 

986 

302 

48  ooi 

°iS 

029 

044 

058 

073 

087 

101 

116 

130 

303 

144 

159 

173 

187 

202 

216 

230 

244 

259 

273 

304 

287 

302 

316 

33° 

344 

359 

373 

387 

401 

416 

305 

43° 

444 

458 

473 

487 

5oi 

5i5 

53° 

544 

558 

IS 

306 

572 

586 

601 

615 

629 

643 

657 

671 

686 

700 

X 

a 

1-5 
3-° 

307 

714 

728 

742 

756 

770 

785 

799 

813 

827 

841 

3 

4-5 

308 

%5 

869 

883 

897 

911 

926 

940 

954 

968 

982 

4 
5 

6.0 

7.5 

309 

996 

*OIO 

*024 

*038 

*052 

*o66 

*o8o 

*094 

*io8 

*I22 

6 

9.0 

310 

49  i36 

15° 

164 

178 

192 

206 

220 

234 

248 

262 

I 

IO-S 

3" 

276 

290 

3°4 

3i8 

332 

346 

360 

374 

388 

4O2 

9 

13-5 

312 

4i5 

429 

443 

457 

471 

485 

499 

5J3 

527 

54i 

3i3 

554 

568 

582 

596 

610 

624 

638 

651 

665 

679 

3M 

693 

707 

721 

734 

748 

762 

776 

790 

803 

817 

3i5 

831 

845 

859 

872 

886 

900 

914 

927 

941 

955 

3i6 

969 

982 

996 

*OIO 

*024 

*°37 

*°5* 

*o65 

*079 

*092 

14 

3i7 

50  106 

120 

133 

147 

161 

174 

188 

202 

215 

229 

x 

2 

i  4 

2  8 

3i8 

243 

256 

270 

284 

297 

3" 

325 

338 

352 

365 

3 

4  2 

3i9 

379 

393 

406 

420 

433 

447 

461 

474 

488 

5°i 

4 

5  6 

320 

5*5 

529 

542 

556 

569 

583 

596 

610 

623 

637 

I 

7  ° 
8  4 

321 

651 

664 

678 

691 

705 

718 

732 

745 

759 

772 

I 

9.8 

322 

786 

799 

8i3 

826 

840 

853 

866 

880 

893 

907 

o 

9 

II  .  2 

12.6 

323 

920 

934 

947 

961 

974 

987 

*OOI 

*oi4 

*028 

*o4i 

324 

5i  °55 

068 

081 

°95 

108 

121 

135 

148 

162 

175 

325 

1  88 

202 

215 

228 

242 

255 

268 

282 

295 

308 

326 

322 

335 

348 

362 

375 

388 

402 

415 

428 

441 

327 

455 

468 

481 

495 

508 

S2! 

534 

548 

561 

574 

13 

328 

587 

601 

614 

627 

640 

654 

667 

680 

693 

706 

i 

i-3 

_   /C 

329 

720 

733 

746 

759 

772 

786 

799 

812 

825 

838 

2 

3 

2  .  O 

3-9 

330 

%i 

865 

'878 

891 

904 

917 

93° 

943 

957 

970 

4 

6  •; 

33i 

983 

996 

*oo9 

*022 

*°35 

*048 

*o6i 

1=075 

*o88 

*IOI 

? 

s 

7.8 

332 

52  114 

127 

140 

153 

1  66 

179 

192 

205 

218 

231 

7 

9.1 

333 

244 

257 

270 

284 

297 

310 

323 

336 

349 

362 

8 
9 

10.4 
II.  7 

334 

375 

388 

401 

414 

427 

440 

453 

466 

479 

492 

335 

5°4 

5i7 

53° 

543 

556 

569 

582 

595 

608 

621 

336 

634 

647 

660 

673 

686 

699 

711 

724 

737 

75° 

337 

763 

776 

789 

802 

8i5 

827 

840 

853 

866 

879 

338 

892 

9°5 

917 

93° 

943 

956 

969 

982 

994 

*oo7 

12 

339 

53  °2o 

°33 

046 

058 

071 

084 

097 

no 

122 

i35 

X 

1.2 

340 

148 

161 

173 

186 

199 

212 

224 

237 

250 

263 

2 

3 

2'1 
3.6 

34i 

275 

288 

301 

3i4 

326 

339 

352 

364 

377 

39° 

4 

4.8 

342 

403 

4i5 

428 

441 

453 

466 

479 

491 

504 

5i7 

! 

6.0 

7.  2 

343 

529 

542 

555 

567 

58o 

593 

605 

618 

631 

643 

7 

8.4 

344 

656 

668 

681 

694 

706 

719 

732 

744 

757 

769 

8 

9.6 
10.8 

345 

782 

794 

807 

820 

832 

845 

857 

870 

882 

895 

346 

908 

920 

933 

945 

958 

970 

983 

995 

*oo8 

*O2O 

347 

54  033 

045 

058 

070 

083 

°95 

108 

120 

133 

145 

348 

158 

170 

183 

195 

208 

220 

233 

245 

258 

270 

349 

283 

295 

3°7 

320 

332 

345 

357 

37° 

382 

394 

156  COMPRESSED  AIR 

TABLE  XIV.  Continued, — LOGARITHMS  OF  NUMBERS 


No. 

o 

I 

2 

3 

4 

5 

6 

7 

8 

9 

Pp.  Pts. 

350 

54  407 

419 

432 

444 

456 

469 

481 

494 

506 

5i8 

351 

53i 

543 

555 

568 

580 

593 

605 

617 

630 

642 

352 

654 

667 

679 

691 

704 

716 

728 

74i 

753 

765 

353 

777 

79° 

802 

814 

827 

839 

851 

864 

876 

888 

354 

900 

9i3 

925 

937 

949 

962 

974 

986 

998 

*on 

355 

55  023 

035 

047 

060 

072 

084 

096 

1  08 

121 

133 

13 

356 

145 

157 

169 

182 

194 

206 

218 

230 

242 

255 

i 

2 

i-3 

2.6 

357 

267 

279 

291 

3°3 

3i5 

328 

340 

352 

364 

376 

3 

3-9 

358 

388 

400 

4i3 

425 

437 

449 

461 

473 

485 

497 

4 

m 

5-2 

6  <; 

359 

5°9 

522 

534 

546 

553 

57° 

582 

594 

606 

618 

6 

W«  5 

7.8 

360 

630 

642 

654 

666 

678 

691 

7°3 

7i5 

727 

739 

7 

9.1 

36i 

75i 

763 

775 

787 

799 

811 

823 

835 

847 

859 

9 

10.4 
11.7 

362 

871 

883 

895 

907 

919 

93i 

943 

955 

967 

979 

363 

991 

*oo3 

*oi5 

*O27 

*o38 

*o5o 

*062 

*o74 

*o86 

+098 

364 

56  no 

122 

134 

146 

158 

170 

182 

194 

205 

217 

365 

229 

241 

253 

265 

277 

289 

301 

312 

324 

336 

366 

348 

360 

372 

384 

396 

407 

419 

43i 

443 

455 

12 

367 

467 

478 

490 

502 

5J4 

526 

538 

549 

56i 

573 

X 

2 

1.2 

2  A 

368 

585 

597 

608 

620 

632 

644 

656 

667 

679 

691 

3 

£  .  4 

3-6 

369 

7°3 

7i4 

726 

738 

75°. 

761 

773 

785 

797 

808 

4 

4-8 
f.  _ 

370 

820 

832 

844 

855 

867 

879 

891 

902 

914 

926 

1 

o  .  o 

7-2 

37i 

937 

949 

961 

972 

984 

996 

*oo8 

*oi9 

*o3i 

*043 

I 

8.4 

_  /c 

372 

57  °54 

066 

078 

089 

101 

i*3 

124 

136 

148 

159 

o 
g 

Q.  O 

10.8 

373 

171 

183 

194 

206 

217 

229 

241 

252 

264 

276 

374 

287 

299 

310 

322 

334 

345 

357 

368 

380 

392 

375 

403 

4i5 

426 

438 

449 

461 

473 

484 

496 

5°7 

376 

5i9 

53° 

542 

553 

565 

576 

588 

600 

611 

623 

377 

634 

646 

657 

669 

680 

692 

7°3 

715 

726 

738 

II 

378 

749 

761 

772 

784 

795 

807 

818 

830 

841 

852 

X 

i.i 

379 

864 

875 

887 

898 

910 

921 

933 

944 

955 

967 

2 

3 

2.  2 

3-3 

380 

978 

990 

*OOI 

*oi3 

*024 

*o35 

*047 

*os8 

*o7o 

*o8i 

4 

4-4 

38i 

58  092 

104 

115 

127 

138 

149 

161 

172 

184 

195 

I 

5  •  5 
6.6 

382 

206 

218 

229 

240 

252 

263 

274 

286 

297 

3°9 

I 

7-7 

383 

320 

33i 

343 

354 

365 

377 

388 

399 

410 

422 

o 
9 

8.  8 
o.o 

384 

433 

444 

456 

467 

478 

490 

5°i 

Si2 

524 

535 

385 

546 

557 

569 

580 

591 

602 

614 

625 

636 

647 

386 

659 

670 

681 

692 

704 

7i5 

726 

737 

749 

760 

387 

771 

782 

794 

805 

816 

827 

838 

850 

861 

872 

388 

883 

894 

906 

917 

928 

939 

95° 

961 

973 

984 

10 

389 

995 

*oo6 

*oi7 

*028 

*040 

*°5i 

*062 

*Q73 

*o84 

*cx)5 

i 

1.0 

390 

59  106 

118 

129 

140 

151 

162 

173 

184 

195 

207 

2 

3 

3-  o 

39i 

218 

229 

240 

251 

262 

273 

284 

295 

306 

3i8 

4 

4.0 

392 

329 

340 

35i 

362 

373 

384 

395 

406 

417 

428 

I 

5-o 
6.0 

393 

439 

45° 

461 

472 

483 

494 

506 

5i7 

528 

539 

I 

7.0 

394 

550 

56i 

572 

583 

594 

605 

616 

627 

638 

649 

8 

8.0 

395 

660 

671 

682 

693 

704 

7i5 

726 

737 

748 

759 

9 

9.0 

396 

770 

780 

791 

802 

813 

824 

835 

846 

857 

868 

397 

879 

890 

901 

912 

923 

934 

945 

956 

966 

977 

398 

988 

999 

*OIO 

*O2I 

*032 

*043 

*054 

*o65 

*o76 

*o86 

399 

60  097 

108 

119 

I30 

141 

152 

163 

173 

184 

195 

TABLES 
•  TABLE  XIV.  Continued. — LOGARITHMS  OF  NUMBERS 


157 


No. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

Pp.  Pts. 

400 

60  206 

217 

228 

239 

249 

260 

271 

282 

293 

3°4 

401 

3i4 

325 

336 

347 

358 

369 

379 

39° 

401 

412 

402 

423 

433 

444 

455 

466 

477 

487 

498 

5°9 

520 

403 

53i 

54i 

552 

563 

574 

584 

595 

606 

617 

627 

404 

638 

649 

660 

670 

68  1 

692 

7°3 

7i3 

724 

735 

405 

746 

756 

767 

778 

788 

799 

810 

821 

831 

842 

406 

853 

863 

874 

885 

895 

906 

917 

927 

938 

949 

407 

959 

97° 

981 

991 

*002 

*OI3 

*023 

*°34 

*045 

°55 

408 

6  1  066 

077 

087 

098 

109 

119 

130 

140 

151 

162 

I 

ii 

i.i 

409 

172 

183 

194 

204 

215 

225 

236 

247 

257 

268 

2 

2.2 

410 

278 

289 

300 

310 

32I 

331 

342 

352 

363 

374 

3 

3-3 

4   A 

411 

384 

395 

405 

416 

426 

437 

448 

458 

469 

479 

5 

•  4 

5-5 

412 

49° 

500 

5" 

52i 

532 

542 

553 

563 

574 

584 

6 

*9 

6.6 

*7  T 

413 

595 

606 

616 

627 

637 

648 

658 

669 

679 

690 

8 

8*.  8 

414 

700 

711 

721 

731 

742 

752 

763 

773 

784 

794 

9 

9.9 

415 

805 

815 

826 

836 

847 

857 

868 

878 

888 

899 

4l6 

909 

920 

93° 

941 

951 

962 

972 

982 

993 

*oo3 

417 

62  014 

024 

034 

045 

055 

066 

076 

086 

097 

107 

4l8 

118 

128 

138 

149 

159 

170 

180 

190 

2OI 

211 

419 

221 

232 

242 

252 

263 

273 

284 

294 

3°4 

315 

420 

325 

335 

346 

356 

366 

377 

38V 

397 

408 

4l8 

42I 

428 

439 

449 

459 

469 

480 

49° 

500 

5" 

S2! 

10 

422 

531 

542 

552 

562 

572 

583 

593 

603 

613 

624 

I 

I.O 

423 

634 

644 

655 

665 

675 

685 

696 

706 

716 

726 

a 

2.0 

424 

737 

747 

757 

767 

778 

788 

798 

808 

818 

829 

3 

4 

3-0 

4.0 

425 

839 

849 

859 

870 

880 

890 

900 

910 

921 

93  1 

5 

l'° 

426 

941 

95i 

961 

972 

982 

992 

*OO2 

*OI2 

*022 

*°33 

o 

7 

6.0 
7.0 

427 

63  043 

•053 

063 

073 

083 

094 

IO4 

114 

124 

134 

8 

8.0 

428 

144 

155 

165 

175 

185 

195 

205 

215 

225 

236 

9 

9.0 

429 

246 

256 

266 

276 

286 

296 

306 

31? 

327 

337 

43<> 

347 

357 

367 

377 

387 

397 

407 

417 

428 

438 

431 

448 

45s 

468 

478 

488 

498 

508 

5i8 

528 

538 

432 

548 

558 

568 

579 

589 

599 

609 

619 

629 

639 

433 

649 

659 

669 

679 

689 

699 

709 

719 

729 

739 

434 

749 

759 

769 

779 

789 

809 

819 

829 

839 

435 

849 

859 

869 

879 

889 

899 

909 

919 

929 

939 

9- 

436 

949 

959 

969 

979 

988 

998 

*oo8 

*oi8 

*028 

*o38 

i 

0.9 

437 

64  048 

058 

068 

078 

088 

098 

108 

118 

128 

137 

2 

1.8 

438 

147 

157 

167 

177 

187 

197 

207 

217 

227 

237 

3 

4 

1:1 

439 

246 

256 

266 

276 

286 

296 

306 

3i6 

326 

335 

I 

4-5 

440 

345 

355 

365 

375 

385 

395 

404 

414 

424 

434 

O 

7 

5-  4 
6.3 

441 

444 

454 

464 

473 

483 

493 

5°3 

5J3 

523 

532 

8 

7-2 

442 

542 

SS2 

562 

572 

582 

59i 

601 

611 

621 

631 

9 

8.1 

443 

640 

650 

660 

670 

680 

689 

699 

709 

719 

729 

444 

738 

748 

758 

768 

777 

787 

797 

807 

816 

826 

445 

836 

846 

856 

865 

875 

885 

895 

904 

914 

924 

446 

933 

943 

953 

963 

972 

982 

992 

*002 

*OII 

*02I 

447 

65  031 

040 

050 

060 

070 

079 

089 

099 

108 

118 

448 

128 

137 

147 

157 

167 

176 

186 

I96 

205 

215 

449 

<?25 

234 

244 

254 

263 

273 

283 

292 

302 

312 

158  COMPRESSED  AIR 

TABLE  XIV.  Continued.— LOGARITHMS  OF  NUMBERS 


No. 

o 

I 

2 

3 

4 

5 

6 

7 

8 

9 

Pp.  Pt8. 

450 

65  321 

33i 

341 

35° 

360 

369 

379 

389 

398 

408 

451 

418 

427 

437 

447 

456 

466 

475 

485 

495 

5°4 

452 

514 

523 

533 

543 

552 

562 

581 

600 

453 

610 

619 

629 

639 

648 

658 

667 

677 

686 

696 

454 

706 

715 

725 

734 

744 

753 

763 

772 

782 

792 

455 

801 

811 

820 

830 

839 

849 

858 

868 

877 

887 

456 

896 

906 

916 

925 

935 

944 

954 

963 

973 

982 

457 

992 

001 

on 

020 

030 

039 

049 

058 

068 

077 

458 

66  087 

096 

106 

"5 

124 

134 

143 

153 

162 

172 

x 

10 

i  .0 

459 

181 

191 

200 

2IO 

219 

229 

238 

247 

257 

266 

2 

2.0 

460 

276 

285 

295 

3°4 

3*4 

323 

332 

342 

361 

3 

3-° 

461 

37° 

38o 

389 

398 

408 

417 

427 

436 

445 

455 

4 
5 

4*  o 
5-° 

462 

464 

474 

483 

492 

502 

511 

521 

53° 

539 

549 

6 

6.0 

463 

558 

567 

577 

586 

596 

605 

614 

624 

633 

642 

8 

7  •  ° 
8.0 

464 

652 

661 

671 

680 

689 

699 

708 

717 

727 

736 

9 

9.0 

465 

745 

755 

764 

773 

783 

792 

801 

811 

820 

829 

466 

839 

848 

857 

867 

876 

885 

894 

904 

913 

922 

467 

932 

94i 

95° 

960 

969 

978 

987 

997 

^006 

*oi5 

468 

67  025 

034 

043 

052 

062 

071 

080 

089 

099 

108 

469 

117 

127 

136 

145 

154 

164 

173 

182 

191 

201 

470 

210 

219 

228 

237 

247 

256 

265 

274 

284 

293 

47i 

302 

3" 

321 

33° 

339 

348 

357 

367 

376 

385 

472 

394 

403 

422 

440 

449 

459 

468 

477 

I 

o.  9 

473 

486 

495 

504 

514 

523 

S32 

54i 

55° 

560 

569 

2 

1.8 

474 

578 

587 

596 

605 

614 

624 

633 

642 

651 

660 

3 

4 

2-7 

3-6 

669 

679 

688 

697 

706 

715 

724 

733 

742 

752 

5 

4-5 

476 

761 

770 

779 

788 

797 

806 

815 

825 

834 

843 

6 
7 

5-4 
6-3 

477 

852 

861 

870 

879 

888 

897 

906 

916 

*925 

934 

8 

7-2 

478 

943 

952 

961 

97° 

979 

988 

997 

*oo6 

*024 

9 

8.1 

479 

68  034 

043 

052 

061 

070 

079 

088 

097 

1  06 

115 

480 

124 

133 

142 

I51 

1  60 

169 

178 

187 

196 

205 

481 

215 

224 

233 

242 

251 

260 

269 

278 

287 

296 

482 

3°5 

3*4 

323 

332 

35° 

359 

368 

377 

386 

483 

395 

404 

413 

422 

43  1 

440 

449 

458 

467 

476 

484 

485 

494 

502 

511 

520 

529 

538 

547 

556 

565 

485 

574 

583 

592 

601 

610 

619 

628 

637 

646 

655 

g 

486 

664 

673 

681 

690 

699 

708 

717 

726 

735 

744 

I 

0.8 

487 

753 

762 

771 

780 

789 

797 

806 

815 

824 

833 

2 

1.6 

488 

842 

851 

860 

869 

878 

886 

895 

904 

9*3 

922 

3 

4 

2.  4 
3-2 

489 

931 

940 

949 

958 

966 

975 

984 

993 

*OO2 

*OII 

1:1 

490 

69  020 

028 

037 

046 

055 

064 

°73 

082 

090 

099 

7 

5:* 

491 

108 

117 

126 

135 

144 

152 

161 

170 

179 

1  88 

8 

6.4 

492 

197 

205 

214 

223 

232 

241 

249 

258 

267 

276 

9 

7.2 

493 

285 

294 

302 

311 

320 

329 

338 

346 

355 

364 

494 

373 

381 

39° 

399 

408 

425 

434 

443 

452 

495 

461 

469 

478 

487 

496 

504 

513 

522 

531 

539 

496 

548 

557 

566 

574 

583 

592 

601 

609 

618 

627 

497 

636 

644 

653 

662 

671 

679 

688 

697 

7°5 

714 

498 

723 

732 

740 

749 

758 

767 

775 

784 

793 

801 

499 

810 

819 

827 

836 

845 

854 

862 

871 

880 

888 

TABLES  159 

TABLE  XIV.  Continued. — LOGARITHMS  or  NUMBERS 


No. 

o 

I 

2 

3 

4 

5 

6 

7 

8 

9 

Pp.  Pt«. 

500 

69  897 

906 

914 

923 

932 

940 

949 

958 

966 

975 

501 

984 

992 

001 

010 

018 

027 

036 

044 

*°53 

062 

502 

70  070 

079 

088 

096 

105 

114 

122 

I3I 

140 

148 

503 

157 

165 

174 

183 

191 

200 

209 

2!7 

226 

234 

504 

243 

252 

260 

269 

278 

286 

295 

3°3 

312 

321 

505 

329 

338 

346 

355 

364 

372 

389 

398 

406 

506 

4J5 

424 

432 

441 

449 

458 

467 

475 

484 

492 

5°7 

501 

5°9 

526 

535 

544 

569 

578 

508 

586 

595 

603 

612 

621 

629 

638 

646 

655 

663 

x 

9 

0.9 

509 

672 

680 

689 

697 

706 

714 

723 

731 

740 

749 

2 

1.8 

5io 

757 

766 

774 

783 

791 

800 

808 

817 

825 

834 

3 

2.7 
i  6 

5" 

842 

851 

859 

868 

876 

885 

893 

902 

910 

919 

4 
5 

3  -° 
4-5 

512 

927 

935 

944 

952 

961 

969 

978 

986 

995 

+003 

6 

5-4 

71  012 

020 

029 

037 

046 

054 

063 

071 

079 

088 

I 

•  3 

7-2 

5J4 

096 

105 

H3 

122 

130 

139 

147 

155 

164 

172 

9 

8.1 

515 

181 

l89 

198 

206 

214 

223 

231 

240 

248 

257 

516 

265 

273 

282 

290 

299 

3°7 

324 

332 

517 

349 

357 

366 

374 

383 

399 

408 

416 

425 

518 

433 

441 

45° 

458 

466 

475 

483 

492 

500 

508 

519 

5I7 

525 

533 

542 

55° 

559 

567 

575 

584 

592 

520 

600 

609 

617 

625 

634 

642 

650 

659 

667 

675 

521 

684 

692 

700 

709 

717 

725 

734 

742 

75° 

759 

522 

767 

775 

784 

792 

800 

809 

817 

825 

834 

842 

.  x' 

0.8 

523 

850 

858 

867 

875 

883 

892 

900 

908 

917 

925 

2 

1.6 

524 

933 

941 

95° 

958 

966 

975 

983 

991 

999 

*oo8 

3 

4 

2.4 

3-2 

525 

72  016 

024 

032 

041 

049 

°57 

066 

074 

082 

090 

5 

4.0 

526 

099 

107 

"5 

123 

132 

140 

148 

156 

165 

173 

6 

4.8 

_  A 

527 

181 

189 

198 

206 

214 

222 

230 

239 

247 

255 

I 

5-0 
6.4 

528 

263 

272 

280 

288 

296 

3°4 

321 

329 

337 

9 

7-2 

529 

346 

354 

362 

37° 

378 

387 

395 

403 

411 

419 

530 

428 

436 

444 

452 

460 

469 

477 

485 

493 

501 

509 

526 

534 

542 

55° 

558 

567 

575 

583 

532 

599 

607 

616 

624 

632 

640 

648 

656 

665 

533 

673 

681 

689 

697 

7°5 

713 

722 

73° 

738 

746 

534 

754 

76* 

770 

779 

787 

795 

803 

8n 

819 

827 

535 

835 

843 

852 

860 

868 

876 

884 

892 

900 

908 

536 

916 

925 

933 

941 

949 

957 

965 

973 

981 

989 

x 

7 
0.7 

537 

997 

*oo6 

*022 

*O3O 

*o38 

*O46 

*°54 

*062 

+070 

2 

i-4 

538 

73  °78 

086 

094 

102 

in 

119 

127 

135 

143 

151 

3 

4 

2.1 
2.8 

539 

159 

167 

175 

183 

191 

199 

207 

215 

223 

231 

i 

3-5 

540 

239 

247 

255 

263 

272 

280 

288 

296 

3°4 

312 

i  O 

fj 

4.2 

320 

328 

336 

344 

352 

360 

368 

376 

384 

392 

I 

si 

542 

400 

408 

416 

424 

432 

440 

448 

456 

464 

472 

9 

6-3 

543 

480 

488 

496 

5°4 

512 

520 

528 

536 

544 

SS2 

544 

560 

568 

576 

584 

592 

600 

608 

616 

624 

632 

545 

640 

648 

656 

664 

672 

679 

687 

695 

703 

711 

546 

719 

727 

735 

743 

751 

759 

767 

775 

783 

791 

547 

799 

807 

815 

823 

830 

838 

846 

854 

862 

870 

548 

878 

886 

894 

902 

910 

918 

926 

*933 

941 

949 

549 

957 

965 

973 

98j 

989 

997 

*oo5 

*02O 

*028 

74 

160  COMPRESSED  AIR 

TABLE  XIV.  Continued. — LOGARITHMS  OF  NUMBERS 


No. 

0 

i 

2 

3 

4 

5 

6 

7 

8 

9 

Pp.  Pts. 

550 

74  036 

044 

052 

060 

068 

076 

084 

092 

099 

107 

551 

"5 

123 

131 

139 

147 

i55 

162 

170 

178 

186 

552 

194 

202 

2IO 

218 

225 

233 

241 

249 

257 

265 

553 

273 

280 

288 

296 

3°4 

312 

320 

327 

335 

343 

554 

359 

367 

374 

382 

390 

398 

406 

414 

421 

555 

429 

437 

445 

453 

461 

468 

476 

484 

492 

500 

556 

5°7 

515 

523 

531 

539 

547 

554 

562 

570 

578 

557 

586 

593 

601 

609 

617 

624 

632 

640 

648 

656 

558 

663 

671 

679 

687 

695 

702 

710 

718 

726 

733 

559 

74i 

749 

757 

764 

772 

780 

788 

796 

803 

811 

560 

819 

827 

834 

842 

850 

858 

865 

873 

881 

889 

896 

904 

912 

920 

927 

935 

943 

95° 

958 

966 

562 

974 

981 

989 

997 

*oo5 

*OI2 

*020 

*028 

*°35 

*°43 

8" 

563 

75  051 

°59 

066 

074 

082 

089 

097 

105 

113 

I2O 

I  o  8 

564 

128 

136 

143 

159 

166 

174 

182 

189 

197 

2  i'.6 

205 

213 

220 

228 

236 

243 

251 

259 

266 

274 

3  2.4 

4  3-2 

566 

282 

289 

297 

3°5 

312 

320 

328 

335 

343 

351 

5  4-0 

567 

358 

366 

374 

381 

389 

397 

404 

412 

420 

427 

6  4.8 
7  c  6 

568 

435 

442 

45° 

458 

465 

473 

481 

488 

496 

5°4 

/   5  •  u 
8  6.4 

569 

511 

5J9 

526 

534 

542 

549 

557 

565 

572 

9  7-2 

570 

587 

595 

603 

610 

618 

626 

633 

641 

648 

656 

57i 

664 

671 

679 

686 

694 

702 

709 

717 

724 

732 

572 
573 

740 
815 

747 
823 

755 
831 

762 
838 

770 
846 

778 
853 

785 
861 

868 

800 
876 

808 

884 

574 

891 

899 

906 

914 

921 

929 

937 

944 

952 

959 

967 

974 

982 

989 

997 

*oo5 

*OI2 

*02O 

*027 

*°35 

576 

76  042 

050 

°57 

065 

072 

080 

087 

°95 

103 

no 

III 

118 
193 

125 
200 

133 
208 

140 
215 

148 
223 

155 
230 

163 
238 

170 
245 

I78 
253 

185 
260 

579 

268 

275 

283 

290 

298 

313 

320 

328 

335 

580 

343 

350 

358 

365 

373 

380 

388 

395 

403 

410 

58i 

418 

425 

433 

440 

448 

455 

462 

470 

477 

485 

7 

582 

492 

500 

5°7 

515 

522 

53° 

537 

545 

SS2 

559 

2    .4 

583 

567 

574 

582 

589 

597 

604 

612 

619 

626 

634 

4   '.8 

584 

641 

649 

656 

664 

671 

678 

686 

693 

701 

708 

5   -5 

585 

716 

723 

73° 

738 

745 

753 

760 

768 

775 

782 

6    .2 

586 

790 

797 

805 

812 

819 

827 

834 

842 

849 

856 

7   •  9 
8  5.6 

587 

864 

871 

879 

886 

893 

901 

908 

916 

923 

93° 

9  6.3 

588 

938 

945 

953 

960 

967 

975 

982 

989 

997 

589 

77  012 

019 

026 

034 

041 

048 

056 

063 

070 

078 

590 

085 

093 

100 

107 

115 

122 

129 

137 

144 

151 

59i 

159 

1  66 

173 

181 

188 

195 

203 

2IO 

217 

'225 

592 

232 

240 

247 

254 

262 

269 

276 

283 

291 

298 

593 
594 

3°5 
379 

386 

320 
393 

327 
401 

3S 

342 
415 

349 
422 

357 
43° 

364 
437 

37i 
444 

595 

452 

459 

466 

474 

481 

488 

495 

503 

510 

517 

596 

525 

532 

539 

546 

554 

561 

568 

576 

583 

59° 

597 

597 

605 

612 

619 

627 

634 

641 

648 

656 

663 

598 

670 

677 

685 

692 

699 

706 

714 

721 

728 

735 

599 

743 

75° 

757 

764 

772 

779 

786 

793 

801 

808 

TABLES 
TABLE  XIV.  Continued. — LOGARITHMS  OF  NUMBERS 


161 


No. 

0 

i 

2 

3 

4 

5 

6 

7 

8 

9 

Pp.  Pts. 

600 

77  815 

822 

830 

837 

844 

851 

859 

866 

873 

880 

601 

887 

895 

902 

909 

916 

924 

93i 

93s 

945 

952 

602 

960 

967 

974 

981 

988 

996 

*oo3 

*OIO 

*oi7 

*025 

603 

78  032 

°39 

046 

°53 

06  1 

068 

075 

082 

089 

097 

604 

104 

in 

118 

125 

132 

140 

147 

154 

161 

1  68 

605 

176 

183 

190 

197 

204 

211 

219 

226 

233 

240 

606 

247 

254 

262 

269 

276 

283 

290 

297 

3°5 

312 

607 

3*9 

326 

333 

340 

347 

355 

362 

369 

376 

383 

608 

39° 

398 

405 

412 

419 

426 

433 

440 

447 

455 

I 

8 

0.8 

609 

462 

469 

476 

483 

49° 

497 

5°4 

512 

519 

526 

2 

1.6 

610 

533 

540 

547 

554 

56i 

569 

576 

583 

59° 

597 

3 

2.4 

611 

604 

611 

618 

625 

633 

640 

647 

654 

661 

668 

4 
5 

3-2 
4.0 

612 

675 

682 

689 

696 

704 

711 

718 

725 

732 

739 

6 

4.8 

613 

746 

753 

760 

767 

774. 

781 

789 

796 

803 

810 

7 
g 

5.6 

6.  4 

614 

817 

824 

831 

838 

845 

852 

859 

866 

873 

880 

9 

7.2 

615 

888 

895 

902 

909 

916 

923 

93° 

937 

944 

95i 

616 

958 

965 

972 

979 

986 

993 

*000 

*oo7 

*oi4 

*02I 

617 

79  029 

036 

043 

050 

°57 

064 

07! 

078 

085 

092 

618 

099 

1  06 

"3 

I2O 

127 

134 

141 

148 

155 

162 

619 

169 

176 

183 

190 

197 

204 

211 

218 

225 

232 

620 

239 

246 

253 

260 

•267 

274 

28l 

288 

295 

302 

621 

3°9 

316 

323 

33° 

337 

344 

351 

358 

365 

372 

622 

379 

386 

393 

400 

407 

414 

421 

428 

435 

442 

I 

7 

O.  7 

623 

449 

456 

463 

47° 

477 

484 

49  1 

498 

5°5 

5" 

2 

i-4 

524 

5i8 

525 

S32 

539 

546 

553 

56o 

567 

574 

58i 

3 

2.  I 
2.8 

625 

588 

595 

602 

609 

616 

623 

630 

637 

644 

650 

5 

3-5 

626 

657 

664 

671 

678 

685 

692 

699 

706 

7i3 

720 

6 

4.2 

627 

727 

734 

74i 

748 

754 

761 

768 

775 

782 

789 

87 

4-9 
5-6 

628 

796 

803 

810 

817 

824 

831 

837 

844 

851 

858 

9 

6.3 

629 

865 

872 

879 

886 

893 

900 

906 

913 

920 

927 

630 

934 

94i 

948 

955 

962 

969 

975 

982 

989 

996 

631 

80  003 

010 

017 

024 

030 

037 

044 

051 

058 

065 

632 

072 

079 

085 

092 

099 

1  06 

H3 

120 

127 

J34 

633 

140 

147 

154 

161 

1  68 

175 

182 

188 

195 

202 

634 

209 

216 

223 

229 

236 

243 

250 

257 

264 

271 

635 

277 

284 

291 

298 

305 

312 

3i8 

325 

332 

339 

5 

636 

346 

353 

359 

366 

373 

380 

387 

393 

400 

407 

I 

0.6 

637 

414 

421 

428 

434 

441 

448 

455 

462 

468 

475 

2 

1.2 

638 

482 

489 

496 

502 

5°9 

5i6 

523 

53° 

536 

543 

3 

4 

1.8 
2.4 

639 

55° 

557 

564 

57° 

577 

584 

59i 

598 

604 

611 

3-0 

640 

618 

625 

632 

638 

645 

652 

659 

665 

672 

679 

6 

rj 

3-6 
4  2 

641 

686 

693 

699 

706 

7*3 

720 

726 

733 

740 

747 

8 

4^8 

642 

754 

760 

767 

774 

78i 

787 

794 

801 

808 

814 

9 

5-4 

643 

821 

828 

835 

841 

848 

855 

862 

868 

875 

882 

644 

889 

895 

902 

909 

916 

922 

929 

936 

943 

949 

645 

956 

963 

969 

976 

983 

990 

996 

*oo3 

*OIO 

*oi7 

646 

81  023 

030 

°37 

043 

050 

°57 

064 

070 

077 

084 

647 

090 

097 

104 

in 

117 

124 

131 

*37 

144 

151 

648 

158 

164 

171 

178 

184 

191 

198 

204 

211 

218 

649 

224 

231 

238 

245 

251 

258 

265 

271 

278 

28S 

11 


162  COMPRESSED  AIR 

TABLE  XIV.  Continued. — LOGARITHMS  OF  NUMBERS 


No. 

o 

i 

2 

3 

4 

5 

6 

7 

8 

9 

Pp.  Pts. 

650 

81  291 

298 

3°5 

311 

3i8 

325 

33i 

338 

345 

35i 

651 

358 

365 

37i 

378 

385 

39i 

398 

405 

411 

418 

652 

425 

43i 

438 

445 

45i 

45s 

465 

471 

478 

485 

653 

491 

498 

5°5 

5ii 

5i8 

525 

53i 

538 

544 

55i 

654 

558 

564 

57i 

578 

584 

59i 

598 

604 

611 

617 

655 

624 

631 

637 

644 

651 

657 

664 

671 

677 

684 

656 

690 

697 

704 

710 

717 

723 

73° 

737 

743 

75o 

657 

757 

763 

770 

776 

783 

79° 

796 

803 

809 

816 

658 

823 

829 

836 

842 

849 

856 

862 

869 

875 

882 

659 

889 

895 

902 

908 

915 

921 

928 

935 

941 

948 

660 

954 

961 

968 

974 

981 

987 

994 

*ooo 

*oo7 

*oi4 

661 

82  020 

027 

°33 

040 

046 

°53 

060 

066 

°73 

079 

662 

086 

092 

099 

i°5 

112 

119 

125 

132 

138 

145 

663 

I51 

158 

164 

171 

I78 

184 

191 

197 

204 

2IO 

7 
I  o.  7 

664 

217 

223 

230 

236 

243 

249 

256 

263 

269 

276 

2  1.4 

665 

282 

289 

295 

302 

308 

3*5 

321 

328 

334 

341 

3  2.1 
4  2.8 

666 

347 

354 

360 

367 

373 

380 

387 

393 

400 

406 

5  3-5 

667 

413 

419 

426 

432 

439 

445 

452 

458 

465 

471 

6  4.2 

668 

478 

484 

49  1 

497 

5°4 

5io 

5i7 

523 

530 

536 

8  J:? 

669 

543 

549 

556 

562 

569 

575 

582 

588 

595 

601 

9  6.3 

670 

607 

614 

620 

627 

633 

640 

646 

653 

659 

666 

671 

672 

679 

685 

692 

698 

7°5 

711 

718 

724 

73° 

672 

737 

743 

750 

756 

763 

769 

776 

782 

789 

795 

673 

802 

808 

814 

821 

827 

834 

840 

847 

853 

860 

674 

866 

872 

879 

885 

892 

898 

9°5 

911 

918 

924 

675 

93° 

937 

943 

95° 

956 

963 

969 

975 

982 

988 

676 

995 

*OOI 

*oo8 

*oi4 

*O2O 

*027 

*Q33 

*040 

*046 

*052 

677 

83  °59 

065 

072 

078 

085 

091 

097 

104 

no 

117 

678 

123 

129 

136 

142 

149 

155 

161 

1  68 

174 

181 

679 

187 

193 

200 

206 

213 

219 

225 

232 

238 

245 

680 

251 

257 

264 

270 

276 

283 

289 

296 

302 

308 

681 

3i5 

321 

327 

334 

340 

347 

353 

359 

366 

372V 

•*?? 
I  0.6 

682 

378 

385 

39i 

398 

404 

410 

4i7 

423 

429 

436 

2   1.2 

683 

442 

448 

455 

461 

467 

474 

480 

487 

493 

499 

3  1.8 
4  2.4 

684 

506 

512 

5i8 

525 

531 

537 

544 

55° 

556 

563 

5  3-0 

685 

569 

575 

582 

588 

594 

601 

607 

613 

620 

626 

6  3.6 

68*6 

632 

639 

645 

651 

658 

664 

670 

677 

683 

689 

I  1:1 

687 

696 

702 

708 

7i5 

721 

727 

734 

740 

746 

753 

9  5-4 

688 

759 

765 

771 

778 

784 

790 

797 

803 

809 

816 

689 

822 

828 

835 

841 

847 

853 

860 

866 

872 

879 

690 

885 

891 

897 

904 

910 

916 

923 

929 

935 

942 

691 

948 

954 

960 

967 

973 

979 

985 

992 

*oo4 

692 

84  on 

017 

023 

029 

036 

042 

048 

055 

061 

067 

693 

°73 

080 

086 

092 

098 

i°5 

in 

117 

123 

130 

694 

136 

142 

148 

155 

161 

167 

i73 

180 

186 

192 

695 

198 

205 

211 

217 

223 

230 

236 

242 

248 

255 

696 

261 

267 

273 

280 

286 

292 

298 

3°5 

3" 

317 

697 

323 

33° 

336 

342 

348 

354 

361 

367 

373 

379 

698 

386 

392 

398 

404 

410 

4i7 

423 

429 

435 

442 

699 

448 

454 

460 

466 

473 

479 

485 

49  1 

497 

504 

TABLES  163 

TABLE  XIV.  Continued. — LOGARITHMS  OF  NUMBERS 


No. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

Pp.  Pt». 

700 

84  510 

5i6 

522 

528 

535 

54i 

547 

553 

559 

566 

701 

S72 

578 

584 

59° 

597 

603 

609 

615 

621 

628 

702 

634 

640 

646 

652 

658 

665 

671 

677 

683 

689 

703 

696 

702 

708 

714 

720 

726 

733 

739 

745 

75i 

704 

757 

763 

770 

776 

782 

788 

794 

800 

807 

813 

705 

8lQ 

825 

831 

837 

844 

850 

856 

862 

868 

874 

706 

880 

887 

893 

899 

9°5 

911 

917 

924 

93° 

936 

707 

942 

948 

954 

960 

967 

973 

979 

985 

991 

997 

708 

85  003 

009 

016 

022 

028 

034 

040 

046 

052 

058 

I 

7 
o.  7 

709 

065 

071 

077 

083 

089 

°95 

101 

107 

114 

I2O 

2 

1.4 

710 

126 

132 

138 

144 

i5° 

156 

163 

169 

175 

181 

3 

2.  I 

711 

187 

i93 

199 

205 

211 

217 

224 

230 

236 

242 

4 
5 

2.  8 

3-5 

712 

248 

254 

260 

266 

272 

278 

285 

291 

297 

3°3 

6 

4-2 

713 

3°9 

3*5 

321 

327 

333 

339 

345 

352 

358 

364 

7 

8 

4.9 
5  6 

7M 

37° 

376 

382 

388 

394 

400 

406 

412 

418 

425 

9 

6.3 

715 

43i 

437 

443 

449 

455 

461 

467 

473 

479 

485 

716 

491 

497 

5°3 

5°9 

5i6 

522 

528 

534 

540 

546 

717 

552 

558 

564 

570 

576 

582 

588 

594 

600 

606 

7l8 

612 

618 

625 

631 

637 

643 

649 

655 

661 

667 

719 

673 

679 

685 

691 

697 

7°3 

709 

7i5 

721 

727 

720 

733 

739 

745 

75i 

757 

763 

769 

775 

781 

788 

721 

794 

800 

806 

812 

818 

824 

830 

836 

842 

848 

722 

854 

860 

866 

872 

878 

884 

890 

896 

902 

908 

j 

o 

0.6 

723 

914 

920 

926 

932 

938 

944 

95° 

956 

962 

968 

a 

1.2 

724 

974 

980 

986 

992 

998 

*oo4 

*OIO 

*oi6 

*O22 

*028 

3 

1.8 

725 

86  034 

040 

046 

052 

058 

064 

070 

076 

082 

088 

• 
-5 

2  .  4 

3-0 

726 

094 

100 

1  06 

112 

118 

124 

130 

136 

141 

147 

6 

3-6 

727 

153 

159 

165 

171 

177 

183 

189 

195 

201 

207 

87 

4-2 
4.8 

728 

213 

219 

225 

23I 

237 

243 

249 

255 

26l 

267 

9 

5-4 

729 

273 

279 

285 

29I 

297 

3°3 

308 

3H 

320 

326 

730 

332 

338 

344 

35° 

356 

362 

368 

374 

380 

386 

731 

392 

398 

404 

410 

4i5 

421 

427 

433 

439 

445 

732 

45i 

457 

463 

469 

475 

481 

487 

493 

499 

5°4 

733 

Si° 

5i6 

522 

528 

534 

540 

546 

552 

558 

564 

734 

570 

576 

58i 

587 

593 

599 

605 

611 

617 

623 

735 

629 

635 

641 

646 

652 

658 

664 

670 

676 

682 

736 

688 

694 

700 

7°5 

711 

717 

723 

729 

735 

74i 

I 

5 

0.5 

737 

747 

753 

759 

764 

77° 

776 

782 

788 

794 

800 

2 

I.O 

738 

806 

812 

817 

823 

829 

835 

841 

847 

853 

859 

3 

4 

i.  5 

2.O 

739 

864 

870 

876 

882 

888 

894 

900 

906 

911 

917 

5 

2-5 

740 

923 

929 

935 

941 

947 

953 

958 

964 

970 

976 

6 

M 

3-o 

74i 

982 

988 

994 

999 

*oo5 

*OII 

*oi7 

*023 

*029 

*°35 

8 

3  •  5 

4.0 

742 

87  040 

046 

052 

058 

064 

070 

075 

081 

087 

093 

9 

4-5 

743 

099 

i°5 

in 

116 

122 

128 

*34 

140 

146 

151 

744 

157 

163 

169 

175 

181 

186 

192 

198 

204 

2IO 

745 
746 

216 

274 

221 
280 

227 
286 

233 
291 

239 

297 

245 
3°3 

251 
3°9 

256 
3*5 

262 
320 

268 
326 

747 

332 

338 

344 

349 

355 

361 

367 

373 

379 

384 

748 

39° 

396 

402 

408 

4i3 

419 

425 

43i 

437 

442 

749 

448 

454 

460 

466 

47i 

477 

483 

489 

495 

500 

164  COMPRESSED  AIR 

TABLE  XIV.  Continued. — LOGARITHMS  OF  NUMBERS 


No. 

0 

i 

2 

3 

4 

5 

6 

7 

8 

9 

Pp.Pt8. 

750 

87  506 

512 

518 

523 

529 

535 

54i 

547 

SS2 

558 

751 

564 

57° 

576 

58i 

587 

593 

599 

604 

610 

616 

752 

622 

628 

633 

639 

645 

651 

656 

662 

668 

674 

753 

679 

685 

691 

697 

703 

708 

714 

720 

726 

73i 

754 

737 

743 

749 

754 

760 

766 

772 

777 

783 

789 

755 

795 

800 

806 

812 

818 

823 

829 

835 

841 

846 

756 

852 

858 

864 

869 

875 

881 

887 

892 

898 

904 

757 

910 

9*5 

921 

927 

933 

938 

944 

95° 

955 

961 

758 

967 

973 

978 

984 

990 

996 

*OOI 

*oo7 

*oi3 

*oi8 

759 

88  024 

030 

036 

041 

047 

053 

058 

064 

070 

076 

760 

081 

087 

°93 

098 

104 

no 

116 

121 

127 

*33 

761 

138 

144 

15° 

156 

161 

167 

173 

I78 

184 

190 

762 

195 

2OI 

207 

213 

218 

224 

230 

235 

241 

247 

763 

252 

258 

264 

270 

275 

281 

287 

292 

298 

3°4 

6 

j  06 

764 

3°9 

315 

321 

326 

332 

338 

343 

349 

355 

360 

2   1.2 

765 

366 

372 

377 

383 

389 

395 

400 

406 

412 

4i7 

3  1.8 

766 

423 

429 

434 

440 

446 

45i 

457 

463 

468 

474 

4  2  •  4 
5  3-o 

767 

480 

485 

491 

497 

502 

508 

5i3 

5i9 

525 

53° 

6  3.6 

768 

536 

542 

547 

553 

559 

564 

57° 

576 

58i 

587 

7   4-2 

8  4-8 

769 

593 

598 

604 

610 

615 

621 

627 

632 

638 

643 

9  5-4 

770 

649 

655 

660 

666 

672 

677 

683 

689 

694 

700 

771 

772 

70S 
762 

711 
767 

717 

773 

722 
779 

728 
784 

734 

790 

739 
795 

Kf 

750 
807 

756 
812 

773 

818 

824 

829 

835 

840 

846 

852 

857 

863 

868 

774 

874 

880 

885 

891 

897 

902 

908 

913 

919 

925 

775 

93° 

936 

94i 

947 

953 

958 

964 

969 

975 

981 

776 

986 

992 

997 

*oo3 

*oo9 

*oi4 

*02O 

*025 

*o3i 

*o37 

777 

89  042 

048 

053 

059 

064 

070 

076 

081 

087 

092 

778 

098 

104 

109 

"5 

I2O 

126 

I3I 

137 

143 

148 

779 

154 

159 

165 

170 

I76 

182 

187 

i93 

198 

204 

780 

209 

215 

221 

226 

232 

237 

243 

248 

254 

260 

78i 

265 

271 

276 

282 

287 

293 

298 

3°4 

310 

3i5 

5 

Io  r 

782 

321 

326 

332 

337 

343 

348 

354 

360 

365 

37i 

u-  j 
2   1.0 

783 

376 

382 

387 

393 

398 

404 

409 

4i5 

421 

426 

3  i.  5 

784 

432 

437 

443 

448 

454 

459 

465 

47° 

476 

481 

5  2-5 

785 

487 

492 

498 

5°4 

5°9 

5*5 

520 

526 

53i 

537 

6  3.0 

786 

542 

548 

553 

559 

564 

57o 

575 

58i 

586 

592 

7  3-5 

o  4.0 

787 

597 

603 

609 

614 

620 

625 

631 

636 

642 

647 

9  4-5 

788 

653 

658 

664 

669 

675 

680 

686 

691 

697 

702 

789 

708 

713 

719 

724 

73° 

735 

74i 

746 

752 

757 

790 

763 

768 

774 

779 

785 

79° 

-  796 

801 

807 

812 

791 

818 

823 

829 

834 

840 

845 

85i 

856 

862 

867 

792 

873 

878 

883 

889 

894 

900 

9°5 

911 

916 

922 

793 

927 

933 

938 

944 

949 

955 

960 

966 

971 

977 

794 

982 

988 

993 

998 

*oo4 

*oo9 

*oi5 

*020 

*026 

*°3i 

795 

90  037 

042 

048 

°53 

059 

064 

069 

°75 

080 

086 

796 

091 

097 

102 

108 

H3 

119 

124 

129 

135 

140 

797 

146 

151 

157 

162 

1  68 

J73 

179 

184 

189 

195 

798 

200 

206 

211 

217 

222 

227 

233 

238 

244 

249 

799 

255 

260 

266 

271 

276 

282 

287 

293 

298 

304 

TABLES  165 

TABLE  XIV.  Continued. — LOGARITHMS  OF  NUMBERS 


No. 

0 

i 

2 

3 

4 

5 

6 

7 

8 

9 

Pp.Pts. 

800 

90  309 

3M 

320 

325 

33i 

336 

342 

347 

352 

358 

801 

363 

369 

374 

380 

385 

39° 

396 

401 

407 

412 

802 

4i7 

423 

428 

434 

439 

445 

450 

455 

461 

466 

803 

472 

477 

482 

488 

493 

499 

504 

509 

5T5 

520 

804 

526 

53i 

536 

542 

547 

553 

558 

563 

569 

574 

805 

580 

585 

59° 

596 

601 

607 

612 

617 

623 

628 

806 

634 

639 

644 

650 

655 

660 

666 

671 

677 

682 

807 

687 

693 

698 

703 

709 

7i4 

720 

725 

73° 

736 

808 

74i 

747 

752 

757 

763 

768 

773 

779 

784 

789 

809 

795 

800 

806 

8n 

816 

822 

827 

832 

838 

843 

810 

849 

854 

859 

865 

870 

875 

881 

886 

891 

897 

811 

902 

907 

913 

918 

924 

929 

934 

940 

945 

95° 

812 

956 

961 

966 

972 

977 

982 

988 

993 

998 

*oo4 

813 

91  009 

014 

020 

025 

030 

036 

041 

046 

052 

°57 

I  o.  6 

814 

062 

068 

°73 

078 

084 

089 

094 

100 

I05 

no 

2   1.2 

815 

116 

121 

126 

132 

137 

142 

148 

153 

158 

164 

3  1.8 
4  2-4 

816 

169 

J74 

1  80 

185 

190 

196 

2OI 

206 

212 

217 

S  3-0 

817 

222 

228 

233 

238 

243 

249 

254 

259 

265 

270 

6  3.6 

818 

275 

281 

286 

291 

297 

302 

3°7 

312 

3^ 

323 

1  J:S 

819 

328 

334 

339 

344 

35° 

355 

360 

365 

371 

376 

9  54 

820 

38l 

387 

392 

397 

403 

408 

4i3 

418 

424 

429 

821 

434 

440 

445 

45° 

455 

461 

466 

47i 

477 

482 

822 

487 

492 

498 

5°3 

508 

5i4 

5*9 

524 

529 

535 

823 

54° 

545 

55i 

556 

56i 

566 

572 

577 

582 

587 

824 

593 

598 

603 

609 

614 

619 

624 

630 

635 

640 

825 

645 

651 

656 

661 

666 

672 

677 

682 

687 

693 

826 

698 

7°3 

709 

7H 

719 

724 

73° 

735 

740 

745 

827 

75i 

756 

761 

766 

772 

777 

782 

787 

793 

798 

828 

803 

808 

814 

819 

824 

829 

834 

840 

845 

850 

829 

855 

861 

866 

871 

876 

882 

887 

892 

897 

903 

830 

908 

9i3 

918 

924 

929 

934 

939 

944 

95° 

955 

831 

960 

965 

971 

976 

981 

986 

991 

997 

*O02 

*oo7 

S 
T  ft  e 

832 

92  OI2 

018 

023 

028 

°33 

038 

044 

049 

054 

°59 

x  0.5 

a  i.o 

833 

065 

070 

°75 

080 

085 

091 

096 

IOI 

106 

in 

3  1.5 

4  2.0 

834 

117 

122 

127 

132 

i37 

143 

148 

153 

158 

163 

5  2.5 

835 

169 

174 

179 

184 

189 

195 

200 

205 

210 

215 

0  3.0 

836 

221 

226 

231 

236 

241 

247 

252 

257 

262 

267 

7  3-5 
8  4.0 

837 

273 

278 

283 

288 

293 

298 

3°4 

3°9 

314 

3i9 

9  4-5 

838 

324 

33° 

335 

340 

345 

35° 

355 

361 

366 

37i 

839 

376 

38l 

387 

392 

397 

402 

407 

412 

4l8 

423 

840 

428 

433 

438 

443 

449 

454 

459 

464 

469 

474 

841 

480 

485 

490 

495 

500 

5°5 

5" 

5i6 

S2! 

526 

842 

531 

536 

542 

547 

SS2 

557 

562 

567 

572 

578 

843 

583 

588 

593 

598 

603 

609 

614 

619 

624 

629 

844 

634 

639 

645 

650 

655 

660 

665 

670 

675 

681 

845 

686 

691 

696 

701 

706 

711 

716 

722 

727 

732 

846 

737 

742 

747 

752 

758 

763 

768 

773 

778 

847 

788 

793 

799 

804 

809 

814 

819 

824 

829 

834 

848 

840 

845 

850 

855 

860 

865 

870 

875 

881 

886 

849 

891 

896 

901 

906 

911 

916 

921 

927 

932 

937 

166  COMPRESSED  AIR 

TABLE  XIV.  Continued. — LOGARITHMS  OF  NUMBERS 


No. 

0 

i 

2 

3 

4 

5 

6 

7 

8 

9 

Pp.  Pts. 

850 

92  942 

947 

952 

957 

962 

967 

973 

978 

983 

988 

851 

993 

998 

*oo3 

*oo8 

*oi3 

*oi8 

*024 

*O29 

*°34 

*o39 

852 

93  °44 

049 

054 

°59 

064 

069 

075 

080 

085 

090 

853 

095 

100 

i°5 

no 

H5 

120 

125 

!3* 

136 

141 

854 

146 

151 

156 

161 

1  66 

171 

176 

181 

186 

192 

855 

197 

202 

207 

212 

217 

222 

227 

232 

237 

242 

856 

247 

252 

258 

263 

268 

273 

278 

283 

288 

293 

857 

298 

3°3 

308 

313 

3i8 

323 

328 

334 

339 

344 

858 

349 

354 

359 

364 

369 

374 

379 

384 

389 

394 

6 

859 

399 

404 

409 

414 

420 

425 

43° 

435 

440 

445 

i 

2 

o.  6 

1.2 

860 

45° 

455 

460 

465 

47° 

475 

480 

485 

490 

495 

3 

1.8 

861 

500 

5°5 

51° 

515 

520 

526 

53i 

536 

54i 

546 

4 
5 

2.4 
3  •  ° 

862 

551 

556 

56i 

566 

571 

576 

58i 

586 

59i 

596 

6 

3-6 

863 

601 

606 

611 

616 

621 

626 

631 

636 

641 

646 

I 

4-3 

A  8 

864 

651 

656 

661 

666 

671 

676 

682 

687 

692 

697 

9 

4  •  o 
5-4 

865 

702 

707 

712 

717 

722 

727 

732 

737 

742 

747 

866 

752 

757 

762 

767 

772 

777 

782 

787 

792 

797 

867 

802 

807 

812 

817 

822 

827 

832 

837 

842 

847 

868 

852 

857 

862 

867 

872 

877 

882 

887 

892 

897 

869 

902 

907 

912 

917 

"922 

927 

932 

937 

942 

947 

870 

952 

957 

962 

967 

972 

977 

982 

987 

992 

997 

871 

94  002 

007 

012 

017 

022 

027 

032 

037 

042 

047 

872 

052 

057 

062 

067 

072 

077 

082 

086 

091 

096 

j 

5 

873 

IOI 

1  06 

III 

116 

121 

126 

131 

136 

141 

146 

2 

^'  5 

I.O 

874 

151 

156 

161 

1  66 

171 

176 

181 

186 

191 

196 

3 

i-5 

875 

2OI 

206 

211 

216 

221 

226 

231 

236 

240 

245 

4 

2.5 

876 

250 

255 

260 

265 

270 

275 

280 

285 

290 

295 

6 

3-0 

877 

300 

3<>5 

3IO 

3i5 

320 

325 

33° 

335 

340 

345 

I 

3-5 
4.0 

878 

349 

354 

359 

364 

3^ 

374 

379 

384 

389 

394 

9 

4-5 

879 

399 

404 

409 

414 

419 

424 

429 

433 

438 

443 

880 

448 

453 

458 

463 

468 

473 

478 

483 

488 

493 

88  1 

498 

5°3 

5°7 

512 

517 

522 

527 

532 

537 

542 

882 

547 

552 

557 

562 

567 

57i 

576 

58i 

586 

59i 

883 

596 

601 

606 

611 

616 

621 

626 

630 

635 

640 

884 

645 

650 

655 

660 

665 

670 

675 

680 

685 

689 

885 

694 

699 

704 

709 

714 

719 

724 

729 

734 

738 

886 

743 

748 

753 

758 

763 

768 

773 

778 

783 

787 

i 

4 

0.4 

887 

792 

797 

802 

807 

812 

817 

822 

827 

832 

836 

2 

0.8 

888 

841 

846 

851 

856 

861 

866 

871 

876 

880 

885 

3 

4 

1.2 

1.6 

889 

890 

895 

900 

905 

910 

9J5 

919 

924 

929 

934 

5 

2.0 

890 

939 

944 

949 

954 

959 

963 

968 

973 

978 

983 

6 

rj 

2.4 
2.8 

891 

988 

993 

998 

*OO2 

*oo7 

*OI2 

*oi7 

*022 

*027 

*032 

I 

3-2 

892 

95  °36 

041 

046 

051 

056 

061 

066 

071 

075 

080 

9 

3-6 

893 

085 

090 

°95 

100 

i°5 

109 

114 

119 

124 

129 

894 

i34 

139 

143 

148 

153 

158 

163 

168 

173 

177 

895 

182 

187 

192 

197 

202 

207 

211 

216 

221 

226 

896 

231 

236 

240 

245 

250 

255 

260 

265 

270 

274 

897 

279 

284 

289 

294 

299 

3°3 

308 

313 

318 

323 

898 

328 

332 

337 

342 

347 

352 

357 

361 

366 

371 

899 

376 

3gi 

386 

39° 

395 

400 

405 

410 

415 

419 

TABLES 
TABLE  XIV.  Continued. — LOGARITHMS  OF  NUMBERS 


167 


No. 

0 

i 

2 

3 

4 

5 

6 

7 

8 

9 

Pp.  Pts. 

goo 

95  424 

429 

434 

439 

444 

448 

453 

458 

463 

468 

901 

472 

477 

482 

487 

492 

497 

5°i 

506 

5" 

5i6 

902 

521 

525 

53° 

535 

540 

545 

55° 

554 

559 

564 

903 

569 

574 

578 

583 

588 

593 

598 

602 

607 

612 

904 

617 

622 

626 

631 

636 

641 

646 

650 

655 

660 

90S 

665 

670 

674 

679 

684 

689 

694 

698 

7°3 

708 

906 

7J3 

718 

722 

727 

732 

737 

742 

746 

75  ! 

756 

907 

761 

766 

77° 

775 

780 

785 

789 

794 

799 

804 

908 

809 

813 

818 

823 

828 

832 

837 

842 

847 

852 

909 

856 

861 

866 

871 

875 

880 

885 

890 

895 

899 

910 

904 

909 

914 

918 

923 

928 

933 

938 

943 

947 

911 

952 

957 

961 

966 

971 

976 

980 

985 

990 

995 

912 

999 

*oo4 

*oo9 

*oi4 

*oi9 

*023 

*028 

*Q33 

*o38 

*O42 

c 

9i3 

96  047 

052 

057 

061 

066 

071 

076 

080 

085 

090 

i  0.5 

914 

095 

099 

104 

109 

114 

118 

123 

128 

133 

137 

2   1.0 

9i5 

142 

147 

152 

156 

161 

166 

171 

175 

180 

185 

3  i.S 
4  2.0 

916 

190 

194 

199 

204 

209 

213 

218 

223 

227 

232 

5  2.5 

917 

237 

242 

246 

251 

256 

261 

265 

270 

275 

280 

o  3.0 

7  3.5 
i  j  j 

918 

284 

289 

294 

298 

3°3 

308 

3J3 

3i7 

322 

327 

o  4.0 

919 

332 

336 

34i 

346 

350 

355 

360 

365 

369 

374 

9  4-5 

920 

379 

384 

388 

393 

398 

402 

407 

412 

4i7 

421 

921 

426 

43i 

435 

440 

445 

45° 

454 

459 

464 

468 

922 

473 

478 

483 

487 

492 

497 

5°i 

506 

5" 

5i5 

923 

520 

525 

53° 

534 

539 

544 

548 

553 

558 

562 

924 

567 

572 

577 

58i 

586 

59i 

595 

600 

605 

609 

925 

614 

619 

624 

628 

633 

638 

642 

647 

652 

656 

926 

661 

666 

670 

675 

680 

685 

689 

694 

699 

7°3 

927 

708 

713 

717 

722 

727 

73i 

736 

74i 

745 

75«> 

928 

755 

759 

764 

769 

774 

778 

783 

788 

792 

797 

929 

802 

806 

811 

816 

820 

825 

830 

834 

839 

844 

930 

848 

853 

858 

862 

867 

872 

876 

881 

886 

890 

93i 

895 

900 

904 

909 

914 

918 

923 

928 

932 

937 

4 

i  0.4 

932 

942 

946 

95i 

956 

960 

965 

97° 

974 

979 

984 

2  0.8 

933 

988 

993 

997 

*002 

*oo7 

*on 

*oi6 

*O2I 

*025 

*030 

3  1.2 
4  1.6 

934 

97  °35 

°39 

044 

049 

°53 

058 

063 

067 

072 

077 

S  2.0 

935 

081 

086 

090 

095 

IOO 

104 

109 

114 

118 

123 

6  2.4 
•7  a  -8 

936 

128 

132 

137 

142 

146 

i5i 

155 

1  60 

165 

169 

7   2.0 

8  3.2 

937 

174 

179 

183 

188 

192 

197 

202 

206 

211 

216 

9  3-6 

938 

220 

225 

230 

234 

239 

243 

248 

253 

257 

262 

939 

267 

271 

276 

280 

285 

290 

294 

299 

3°4 

308 

940 

3*3 

3J7 

322 

327 

331 

336 

340 

345 

35o 

354 

941 

359 

364 

368 

373 

377 

382 

387 

39i 

396 

400 

942 

405 

410 

414 

419 

424 

428 

433 

437 

442 

447 

943 

45i 

456 

460 

465 

470 

474 

479 

483 

488 

493 

944 

497 

502 

506 

5u 

5i6 

520 

525 

529 

534 

539 

945 

543 

548 

SS2 

557 

562 

566 

571 

575 

580 

585 

946 

589 

594 

598 

603 

607 

612 

617 

621 

626 

630 

947 

635 

640 

644 

649 

653 

658 

663 

667 

672 

676 

948 

681 

685 

690 

695 

699 

704 

708 

7i3 

717 

722 

949 

727 

73i 

736 

740 

745 

749 

754 

759 

763 

768 

168  COMPRESSED  AIR 

TABLE  XIV.  Continued. — LOGARITHMS  OF  NUMBERS 


No. 

o 

I 

2 

3 

4 

5 

6 

7 

8 

9 

Pp.PtS. 

950 

97  772 

777 

782 

786 

791 

795 

800 

804 

809 

8i3 

951 

8x8 

823 

827 

832 

836 

841 

845 

850 

855 

859 

952 

864 

868 

873 

877 

882 

886 

891 

896 

900 

9°5 

953 

909 

914 

918 

923 

928 

932 

937 

941 

946 

95° 

954 

955 

959 

964 

968 

973 

978 

982 

987 

991 

996 

955 

98  ooo 

005 

009 

014 

019 

023 

028 

032 

037 

041 

956 

046 

050 

055 

059 

064 

068 

073 

078 

082 

087 

957 

091 

096 

100 

i°5 

109 

114 

118 

123 

127 

132 

958 

137 

141 

146 

15° 

155 

159 

164 

168 

173 

177 

959 

182 

186 

191 

195 

200 

204 

209 

214 

218 

223 

960 

227 

232 

236 

241 

245 

250 

254 

259 

263 

268 

961 

272 

277 

281 

286 

290 

295 

299 

3°4 

308 

3*3 

962 

318 

322 

327 

33i 

336 

340 

345 

349 

354 

358 

963 

363 

367 

372 

376 

381 

385 

390 

394 

399 

403 

5 

i  0.5 

964 

408 

412 

417 

421 

426 

43° 

435 

439 

444 

448 

2   1.0 

965 

453 

457 

462 

466 

47i 

475 

480 

484 

489 

493 

3  i.S 

4  2  .  o 

966 

498 

502 

5°7 

5« 

5i6 

520 

525 

529 

534 

SJ8 

5  2.5 

967 

543 

547 

552 

556 

56i 

565 

57° 

574 

579 

583 

6  3.0 

968 

588 

592 

597 

601 

605 

610 

614 

619 

623 

628 

7  3-5 
8  4.0 

969 

632 

637 

641 

646 

650 

655 

659 

664 

668 

673 

9  4-5 

970 

677 

682 

686 

691 

695 

700 

704 

709 

713 

717 

971 

722 

726 

73  1 

735 

740 

744 

749 

753 

758 

762 

972 

767 

771 

776 

780 

784 

789 

793 

798 

802 

807 

973 

811 

816 

820 

825 

829 

834 

838 

843 

847 

851 

974 

856 

860 

865 

869 

874 

878 

883 

887 

892 

896 

975 

.  900 

9°5 

909 

914 

918 

923 

927 

932 

936 

94i 

976 

945 

949 

954 

958 

963 

967 

972 

976 

981 

985 

977 

989 

994 

998 

*do3 

*oo7 

*OI2 

*oi6 

*02I 

*025 

*029 

978 

99  °34 

038 

043 

047 

052 

056 

06  1 

065 

069 

074 

979 

078 

083 

087 

092 

096 

1OO 

i°5 

109 

114 

118 

980 

123 

127 

I31 

136 

140 

145 

149 

154 

158 

162 

981 

167 

171 

176 

180 

185 

189 

193 

198 

202 

207 

4 

I  0.4 

982 

211 

216 

220 

224 

229 

233 

238 

242 

247 

251 

2  0.8 

983 

255 

260 

264 

269 

273 

277 

282 

286 

29I 

295 

3  1.2 
4  1.6 

984 

300 

3°4 

308 

3*3 

3*7 

322 

326 

33° 

335 

339 

5  2.0 

985 

344 

348 

352 

357 

361 

366 

37° 

374 

379 

383 

6  2.4 

FT   /»  Q 

986 

388 

392 

396 

401 

405 

410 

414 

419 

423 

427 

7  2.8 
8  3-2 

987 

432 

436 

441 

445 

449 

454 

458 

463 

467 

47i 

9  3-6 

988 

476 

480 

484 

489 

493 

498 

502 

506 

511 

515 

989 

520 

524 

528 

533 

537 

542 

546 

55° 

555 

559 

990 

564 

568 

572 

577 

58i 

585 

590 

594 

599 

603 

991 

607 

612 

616 

621 

625 

629 

634 

638 

642 

647 

992 

651 

656 

660 

664 

669 

673 

677 

682 

686 

691 

993 

695 

699 

704 

708 

712 

717 

721 

726 

73° 

734 

994 

739 

743 

747 

752 

756 

760 

765 

769 

774 

778 

995 

782 

787 

791 

795 

800 

804 

808 

813 

817 

822 

996 

826 

830 

835 

839 

843 

848 

852 

856 

861 

865 

997 

870 

874 

878 

883 

887 

891 

896 

900 

904 

909 

998 

9J3 

917 

922 

926 

93° 

935 

939 

944 

948 

952 

999 

957 

961 

965 

970 

974 

978 

983 

987 

991 

996 

APPENDIX  A 

The  following  notes  and  tables  relating  to  drill  capacities  are 
taken  from  the  Ingersoll-Rand  catalog. 

DRILL  CAPACITY  TABLES 

The  following  tables  are  to  determine  the  amount  of  free  air 
required  to  operate  rock  drills  at  various  altitudes  with  air  at 
given  pressures. 

The  tables  have  been  compiled  from  a  review  of  a  wide  expe- 
rience and  from  tests  run  on  drills  of  various  sizes.  They  are  in- 
tended for  fair  conditions  in  ordinary  hard,  rock,  but  owing  to 
varying  conditions  it  is  impossible  to  make  any  guarantee  with- 
out a  full  knowledge  of  existing  conditions. 

In  soft  material  where  the  actual  time  of  drilling  is  short,  more 
drills  can  be  run  with  a  given  sized  compressor  than  when  working 
in  hard  material,  when  the  drills  would  be  working  continuously 
for  a  longer  period,  thereby  increasing  the  chance  of  all  the  drills 
operating  at  the  same  time. 

In  tunnel  work,  where  the  rock  is  hard,  it  has  been  the  expe- 
rience that  more  rapid  progress  has  been  made  when  the  drills 
were  operated  under  a  high  air  pressure,  and  that  it  has  been 
found  profitable  to  provide  compressor  capacity  in  excess  of  the 
requirements  by  about  25  per  cent.  There  is  also  a  distinct  ad- 
vantage in  having  a  compressor  of  large  capacity,  in  that  it  saves 
the  trouble  and  expense  of  moving  the  compressor  as  the  work 
progresses,  and  will  not  interfere  with  the  progress  of  the  work  by 
crowding  the  tunnel. 

No  allowance  has  been  made  in  the  tables  for  loss  due  to  leaky 
pipes,  or  for  transmission  loss  due  to  friction,  but  the  capacities 
given  are  merely  the  displacement  required,  so  that  when  select- 
ing a  compressor  for  the  work  required  these  matters  must  be 
taken  into  account. 

Table  I  gives  cubic  feet  of  free  air  required  to  operate  one  drill 
of  a  given  size  and  under  a  given  pressure. 

Table  II  gives  multiplication  factors  for  altitudes  and  number 

169 


170 


COMPRESSED  AIR 


of  drills  by  which  the  air  consumption  of  one  drill  must  be  multi- 
plied in  order  to  give  the  total  amount  of  air. 

TABLE  1. — CUBIC  FEET  OP  FREE  AIR  REQUIRED  TO  RUN  ONE  DRILL  OF  THE 
SlZE  AND  AT  THE  PRESSURE  STATED  BELOW 


2 

SIZE  AND  CYLINDER  DIAMETER  OF  DRILL 

A35 

A32 

B 

C 

D 

D 

D 

E 

F 

F 

G 

H 

H9 

§ 

o 

2" 

2i" 

2J" 

2|" 

3" 

8*" 

3ft" 

3J" 

3J" 

3f" 

4f' 

5" 

«*" 

60 

50 

60 

68 

82 

90 

95 

97 

100 

108 

113 

130 

150 

164 

70 

56 

68 

77 

93 

102 

108 

110 

113 

124 

129 

147 

170 

181 

80 

63 

76 

86 

104 

114 

120 

123 

127 

131 

143 

164 

190 

207 

90 

70 

84 

95 

115 

126 

133 

136 

141 

152 

159 

182 

210 

230 

100 

77 

92 

104 

126 

138 

146 

149 

154 

166 

174 

199 

240 

252 

GLOBE  VALVES,  TEES  AND  ELBOWS 

The  reduction  of  pressure  produced  by  globe  valves  is  the  same 
as  that  caused  by  the  following  additional  lengths  of  straight  pipe, 
as  calculated  by  the  formula: 

A^-*-       n       j.i.    e    •  114  X  diameter  of  pipe 

Additional  length  of  p.pe  =      i  +  (36  _,  diameter) 

£     2     2^      3       3^       456  inches 


Diameter  of  pipe  1    1     1 

Additional  length  J    2     4  7  10  13  16 

78  10  12  15  18 

44  53  70  88  115  143 


20    28     36  feet 
20     22     24  inches 
162  181  200  feet 


The  reduction  of  pressure  produced  by  elbows  and  tees  is  equal 
to  two-thirds  of  that  caused  by  globe  valves.  The  following  are 
the  additional  lengths  of  straight  pipe  to  be  taken  into  account  for 
elbows  and  tees.  For  globe  valves  multiply  by  %. 


Diameter  of  pipe 

\    1 

1J 

i    2 

2^     3     3fc 

5         4 

5 

6 

inches 

Additional 

length 

I    2 

3 

5 

7 

9 

11 

13 

19 

24 

feet 

7 

8 

'10 

12 

15 

18 

20 

22 

24 

inches 

30 

35 

47 

59 

77 

96 

108 

120 

134 

feet 

These  additional  lengths  of  pipe  for  globe  valves,  elbows  and 
tees  must  be  added  in  each  case  to  the  actual  lengths  of  straight 
pipe.  Thus,  a  6-inch  pipe,  500  ft.  long,  with  1  glove  valve,  2 
elbows  and  3  tees,  would  be  equivalent  to  a  straight  pipe  500  + 
36  +  (2  X  24)  +  (3  X  24)  =  656  feet  long. 


APPENDIX  A 


171 


ESSOR  REQUIRED 
D  WITH  SEA  LEV 


g  2 

«g 

go 

1-    B 

&   o 

§B 

5  S 
o^ 


ETERM 
DRILLS 


8AOqi2 


co  TJ<  oq       oo  <* 


<O  00  »O  t~  -^         COCO         CO  •«*  CO 


rt<  00  <M  CO         1C  (M 


OO'-HOSCOCO 


OO  1^  lO  CO  (M         QO  t-  CO  •<*  (N  QO 


OOOOOOOOOSO5OSO5OOO1-H--I 


COCOOOOCO«Ol>»O5i— ICOl>-i— I 
t^t^t^OOOOOOOOOOOSOSOSO 


O5  lO  IO  i—  I  <—  I          O5OSOOOO 
cOO5r-Hrt<COOOO5r-iCO»O 


cOOO"«*iO(MOO-<t" 


OOO5t^OOCO-*(M 


CO-*(MO 

r-icMco"* 


O^t^O 
'-H'-lr-i<N 


COCOOitMt^CO 


30 


Tab 
72 
ns,  t 


APPENDIX  B 
DESIGN  OF  LOGARITHMIC  COMPUTING  CHARTS 

Problem. — Design  a  chart  for  determining  values  of  x,  y  and  z 
in  equations  of  the  form: 

xn  =  aymzr 
or 

I   mr 

x  =  anynzn 
whence 

logo;  =  -  log  a  +  ^  logy  +  -  log  z  (I) 

71  TV  71 

As  a  preliminary  and  introductory  study  assume  n  =  m  +  r 
and  construct  a  chart  as  follows  (See  Fig.  27) : 

Tabulate  values  of  x,  y  and  z  to  be  covered  or  included  in  the 
chart.  Take  out  the  logarithms  of  these  numbers.  Plat  these 
logarithms,  to  some  convenient  scale,  on  the  vertical  lines  marked 
x,  y  and  z  setting  the  zero  of  scale  at  A  and  B  for  the  y  and  z 
lines  respectively,  but  for  the  x  line  set  the  zero  of  scale  at  F 

making  FG  =  -  log  a.  On  the  lines  x}  y  and  z  mark  the  num- 
bers whose  logs  have  been  scaled.  Then  evidently  wherever  the 
line  CD  may  be  placed  across  the  chart  the  proportions  thereon 
written  will  hold  and  Eq.  (I)  is  completely  satisfied — that  is — 
given  any  two  of  x,  y  and  z  the  third  will  be  found  on  the  line 
CD  laid  over  the  two  given. 

Note  that  the  line  AE  is  unnecessary — it  being  placed  in  the 
figure  for  demonstration  only.  Note  also  that  the  line  FG  is  not 
to  appear  on  the  chart  and  that  the  factor  C  effects  only  the  width 
of  the  chart  and  may  be  taken  to  suit  convenience. 

Evidently  this  solution  applies  only  to  the  special  case  when 
n  =  m  +  r.  It  has  the  further  objection  that  if  the  corre- 
sponding numerical  values  of  x,  y  and  z  are  very  different  then 
the  lengths  of  the  x,  y  and  z  lines  will  be  different,  though  not 
in  the  same  proprtion.  The  chart  will  have  a  better  appearance 
if  the  three  lines  are  nearly  equal. 

The  general  solution  is  as  follows  (See  Fig.  28) : 

172 


APPENDIX  B 


173 


Let 


I   =  desired  length  of  chart  in  inches, 
k    =  desired  width  of  chart  in  inches, 
Xi  =  greatest  value  of  x  to  appear  on  the  chart, 
fx  be  the  necessary  multiplier  for  logs  x, 
to  give  desired  length  to  the  x  line. 


logz 


Then  /*  (log  xi  -  -  log  a)  =  I.     Whence  /, 
Let  z\  =  be  the  value  of  z  corresponding  with 


(ID 


174 


COMPRESSED  AIR 


Let  fz  be  the  nearest  whole  number  to 


I 


(III) 


log  21 

that  being  the  most  convenient  multiplier  for 
logs  of  z  to  make  the  z  line  nearly  equal  to  the 
x  line. 


F-l 


FIG.  28. 


Let  fy  be  the  necessary  multiplier  of  logs  y. 

We  have  yet  to  find  p,  q  and  fy. 

Imposing  the  condition  that  Eq.  (I)  must  be  satisfied  and 
remembering  that  all  values  of  log  x  'are  to  be  multiplied  by  fx. 
Then  must 


APPENDIX  B  175 

£/,  log  z  =  rfx  log  z.     Whence  p  =  -£fc  (IV) 

K  n  w>jz 

also 

f  /,  log  »  =  ^  /.  log  t/.     Whence  /„  =  ^  £  fc  (V) 

n/  /6  11     (^ 

Evidently  q  =  k  -  p  (VI) 

Example. — Design  a  chart  to  solve  the  formula  for  friction  in 
ai/pipes,  viz., 

0.10250*1 


rd5-31  X  3,600 
in  which 

/  =  loss  of  pressure  in  pounds  per  square  inch, 

I  =  length  of  pipe  in  feet, 

v  =  cubic  feet  of  free  air  per  minute, 

r  =  ratio  of  compression  =  number  of  atmospheres, 

d  =  diameter  of  air  pipe  in  inches. 

Here  we  find  five  variables  while  our  chart  can  provide  for  three 
only.  We  will,  therefore,  take  I  =  1,000  feet  and  replace  the 
product  fr  with  a  single  variable  and  represent  it  by  h.  The  equa- 
tion will  now  become 

fr  =  h  =  ^3  ^  or  v*  =  35.13  /id5-31 
Whence 

log  v  =  I  log  35.13  +  -hog  h  +  ^  log  d          (VII) 

A  &  A 

which  is  in  the  same  form  as  Eq.  I. 

We  will  design  the  chart  to  be  about  12  in.  long  (I  =  12)  and 
8  in.  wide  (k  =  8)  and  will  provide  for  a  Max.  v  =  50,000  (=v\) 
log  50,000  =  4.6990  and  log  35.13  =  1.5456 
whence  by  II, 

/„  (4.699  -  1^5?)  =  12, 

\  a       I 

whence  fv  =  3  (nearest  whole  number). 

The  value  of  d  corresponding  to  v  =  50,000  is  about  12  in. 

log  12  =  1.0792.     Whence  by  III,  fd  =  1  Q7Q2  =  12   (nearest 

whole  number). 
Then  by  IVr 

p  =  ~-  x  8  X  T^  =  5.31  and  q  =  8  -  531.  =  2.69 


176 

and  by  V 


COMPRESSED  AIR 


f  h  = 
Jh       2 


2.69 


X  3  =  4.461 


See  Plate  III,  page  53,  for  the  completed  chart. 

To  lay  out  such  a  chart,  make  a  table  such  as  is  indicated  below. 
Then  measure  out,  on  the  respective  lines,  with  a  scale  of  inches 
and  decimals  (engineers  scale)  the  quantities  in  columns  headed 
fv  log  v,  fh  log  h  and  fd  log  d  remembering  that  log  1  =  0.0. 
Hence  the  bottom  of  each  line  A,  F  and  B  will  be  marked  1.  As 
each  multiplied  log  is  marked  on  its  line,  write  there  the  corre- 
sponding number  in  columns  headed  v,  h  and  d  respectively. 

On  a  chart  thus  laid  out  a  thread  stretched  as  at  CD  will  lie 
over  the  three  quantities  that  will  satisfy  the  equation,  hence  any 
two  being  known  the  third  can  be  found. 

TABLE  FOR  CHARTING  EQUATION,  v2  =  35.13  hd5-n 
Note  Y2  log  35.13  =  0.7728  and  3  X  0.7728  =  2.318 


v  line 

h  line 

d  line 

V 

log* 

/.  log  * 

h 

log  h 

h  log  h 

d 

logd 

fd  log  d 

The  following  notes  may  help  the  student  when  designing 
charts  for  other  equations  of  the  form  given  in  Eq.  I. 

(a)  If  the  constant  (a,  Eq.  I)  becomes  a  proper  fraction  its  log. 
is  minus  and  the  point  F  must  be  set  above  G  instead  of  below; 
the  zero  of  scale  to  be  set  at  F  when  measuring  out  the  X  logs. 

(6)  If  the  equation  takes  the  form 


then 


1  -,  m  ,  r  , 

log  x  =  -  log  a  -  -  log  y  -  -  log  z 


APPENDIX  B 


177 


This  can  be  satisfied  by  reversing  the  direction  of  the  measure- 
ments on  the  x  line  and  placing  the  zero  (or  F)  point  above  the 

line  AB  a  distance     log  afx  as  indicated  in  Fig.  29. 

Tl 

(c)  When  values  of  any  one  of  the  variables  must  be  fractional 
the  log  is  minus  and  must  be  measured  in  the  opposite  direction 
from  A,  B  or  F  as  the  case  may  be.  For  instance  if  in  the  ex- 


z 

100  4 
100- 


10- 


"ioo:  G 


10. 


FIG.  29. 


ample  above  d  =  Y±  in.  then  log  d  =  1.8751  =  —0.1249  and  we 
must  set  %  at  0.1249  fd   below -B. 

(d)  If  circumstances  are  such  that  in  the  solution  of  such  prob- 
lems as  occur  in  practice ;  the  figures  on  the  lower  portion  of  one 
of  the  lines  (say  the  y  line)  will  never  be  needed;  the  chart  may 
be  set  out  as  suggested  in  Fig.  30  and  only  that  portion  above  BH 
retained  on  the  finished  chart.  Thus  the  scale  may  "be  enlarged 
and  accuracy  increased  thereby. 
12 


178 


COMPRESSED  AIR 


Evidently  the  essential  proportions  of  Fig.  27  are  not  changed 
by  putting  the  chart  in  the  shape  of  Fig.  30. 


i  F 
FIG.  30. 


(e)  It  will  be  found  convenient  to  let  x,  in  the  above  discus- 
sion, represent  the  largest  factor  in  the  equation. 


APPENDIX  C 

During  1910  and  1911,  an  extensive  series  of  experiments  were 
made  at  Missouri  School  of  Mines  to  determine  the  laws  of  fric- 
tion of  air  in  pipes  under  three  inches  in  diameter;  the  chief  object 
being  to  determine  the  coefficient 

"c"  in  the  formula  /  =  c  j5  —      (See  Art.  29.) 

The  general  scheme  is  illustrated  in  Fig.  31,  in  which  the  parts 
are  lettered  as  follows: 

a,  is  the  compressed-air  supply  pipe. 

b,  a  receiver  of  about  25  cu.  ft.  capacity. 

c,  a  thermometer  set  in  receiver. 

d  and  d,  points  of  attachment  of  differential  gage. 


(i) 
FIG.  31. — Diagram  illustrating  assembled  apparatus. 

/  and  /,  lengths  of  straight  pipe  going  to  and  from  the  group  of 
fittings. 

e,  the  pressure  gage. 

g,  the  group  of  fittings — varied  in  different  experiments. 

h,  the  throttle  valve  to  control  pressure. 

7,  the  orifice  drum  for  measuring  air,  with  the  attachments  as 
in  Fig.  9. 

Experiments  at  Missouri  School  of  Mines — 1911 

On  each  set  of  fittings  there  were  made  ten  or  twelve  runs  with 
varying  pressures  and  quantities  of  air  in  order  to  show  the  rela- 

#a2 

tion  of  /  to  —  over  as  wide  a  field  as  possible. 

179 


180 


COMPRESSED  AIR 


£ 


O      S 
«      O 


i  CO  •**!  O  i-H  <M  i— i  O  »— i  O  O  i-» 

r,  i       uuw«^r-»C<ICOCOl>-l'*i'Tt<O>COC^OOCOcNCO'*tl 
~j  ,  *     rH  '^  tO  CO  T— ^  C^  »™^  CO  C^  CO  CO  '"^  CO  CO  rH  CO  CO  T-H 

TH 

c 

I 

O: 

THT^lOi-ICO^COC^lOT-lfO»OrHCOCOT-(COCO 

C^iMfN^r^^cOcOcOOOOOOOCiOiOSi 

c<i  CM'  IN'  -*'  Tt<  •*  cd  cd  cd  06  06  06  oi  cJ  os 

s 

gg^^s 

I-HCO-^OT-KC^I— iOC<IOOr-iOO'-tOO' 

O 

*9      ^ 
«*a 

Ot^CiT-HCO»OCNT-iTtOi-HCOOr-i<NO 

I 


APPENDIX  C  181 

The  data  of  each  run  were  worked  up  and  recorded  in  tabular 
form.  Three  of  these  tables,  relating  to  1-in.  pipe  and  fittings, 
are  shown  herewith  as  example.  It  should  be  recorded  that  in  the 
series  of  runs  and  checks  some  puzzling  inconsistencies  developed, 
but  not  more  noticeable  than  appears  in  the  data  from  European 
experiments  on  larger  pipe. 

In  these  tables  the  symbols  are  as  follows : 

z  =  head,  in  inches  of  mercury,  in  differential  gage, 
/  =  lost  pressure  in  pounds  per  square  inch, 

P2  =  gage  pressure  at  entrance  to  pipe, 

rm  =  mean  ratio  of  compression  in  pipe, 

i   =  water  head,  in  inches,  in  U  tube  on  orifice  drum, 

Tc  =  temperature  (centigrade)  in  drum, 

d0  =  diameter,  in  inches,  of  orifice  in  drum, 

va  =  volume  of  free  air  passing  (cubic  feet  per  second), 
S  =  velocity  of  compressed  air  in  pipe  (feet  per  second), 

/'  =  value  of  /  when  corrected  for  temperature. 


182 


COMPRESSED  AIR 


a  fi 

!• 


o  'S 
&  § 

83-2 


a 


•s^ 
1^ 

O         T— I 


H     a 

*§  I  *s 

5     PLH      05 

»  §  f 
B  g  ^ 
«  g  -3 

a  3  2 

g  p  a 
I  $-3 

|i£ 

w  ^ 

> 


<N  CO  l>  <N  O  rf<  O  i-H  CO  O  i-  (N  O  O  (N  O  O  ^- 


I>O500OO(N'-il>.CX3cOCDi—  il^(NTtH^O5iO 

Ttl  1C  O  CO  <M  iO  i-l  <M  CO  i-l  (N  <M  r-H  i-H  (N  r-l 


1^»  T—  ^  OO  CO  O^  00  ^H  to  t^-  *O  T^  O^  CC  O^  i~H  Oti  ^D  OO 
OiC^i—  iCOt—  i—  it^-t~-CX)t^-COO»OOOOCSlC5O5 
i—  iCOCO(NO'<tlO'-H<NO'-t<MOOC^OOi—  i 


rfi 

OiCO 

t-i  eo  »o  Tt  ,-  i>  ^- 


lOt^t^u^lOCOCOCOCOOOOOOOOOOCOOCO 


APPENDIX  C 


183 


si 


w    § 
§  § 


fe     <N 

«    ff 

B  I 
S  ? 
34 


%*" 


§ 


OOOO  iO  iO*OO  OOOOOOOOOO 


o  i—  ic^"—  i 

C<l  CO          i—  1 


184 


COMPRESSED  AIR 


On  platting  the  values  of /and  £  as  corresponding  coordinates, 
it  becomes  apparent  that  they  are  related  to  each  other  in  all  cases 
as  ordmates  to  a  straight  line;  which  could  have  been  anticipated 
from  the  estabhshed  laws  of  fluid  frictions.  This  is  shown  on 


Values  of  "£" 

From  this  plate  we  get  the  following  three  equations: 
80.0  #  +  20-1-5^-1-40  =  18.3, 

-2e  +  13m        =    6.8,' 


80.0 


APPENDIX  C  185 

in  which 

Va2 

K—  =  resistance  due  to  one  foot  of  pipe, 

v  2 
e  _£_  __  resistance  due  to  one  elbow, 


v  2 
m—  =  resistance  due  to  one  extra  ferrule  or  joint 

with  ends  reamed, 

v  2 
u—  =  resistance  due  to  one  extra  ferrule  or  joint 

with  ends  unreamed, 


a 

g—-=  resistance  due  to  one  globe  valve. 

So  by  attaching  other  lengths  or  fittings  we  get  other  equations 
and  by  simple  algebra  can  find  the  numerical  value  of  each  symbol. 
Then 

.,   2  7     V   2 

Kl~  =  c~—     or    c  =  d5K. 
r          db    r 

Also  the  length  of  pipe  giving  friction  equal  to  that  of  one  elbow 

/> 

is  r,  and  so  with  other  fittings. 

These  experiments  covered  standard  galvanized  pipes  of  2,  1J£, 
1,  %,  and  J^-inch  diameter.  With  each  size  pipe,  runs  were  made 
to  find  friction  loss  in  ordinary  elbows,  45°  elbows,  globe  valves, 
return  bends,  unreamed  joints,  and  reamed  joints.  For  each 
combination,  data  were  taken  for  platting  twelve  to  eighteen 
points,  altogether  about  eight  hundred.  The  results  as  a  whole 
are  satisfactory  for  the  2-,  1J^-,  and  1-inch  pipes. 

For  the  %-  and  J^-inch  pipes,  especially  the  H~m  pipe,  the 
results  were  so  irregular,  erratic,  and  conflicting  that  the  results 
finally  recorded  cannot  be  accepted  as  final.  In  the  light  of  these 
results,  it  is  not  probable  that  a  satisfactory  coefficient  will  ever 
be  gotten  for  pipes  under  1  inch  ;  the  reason  being  that  in  pipes  of 
such  small  diameter,  irregularities  have  relatively  much  greater 
effect  than  in  larger  pipes,  and  the  probability  of  obstructions 
lodging  in  such  pipes  is  relatively  greater.  In  the  J^-inch  pipe 
and  fitting,  unreamed  joints  were  found  at  which  four-tenths  of 
the  area  was  obstructed  and  this  with  a  knife  edge.  No  doubt 
consistent  results  could  have  been  gotten  by  using  only  pipes 
that  had  been  "  plugged  and  reamed,"  and  selected  fittings,  but 
these  results  would  not  have  been  a  safe  guide  for  practice  unless 
such  preparation  of  the  pipe  be  specified. 


186 


COMPRESSED  AIR 


The  results  of  these  researches  are  embodied  in  Plate  II.  They 
show  the  averages  of  such  data  as  seem  worthy  of  consideration. 
The  data  for  pipes  exceeding  2-in.  diameter  are  taken  from 
various  published  data.  Verification  of  these  by  the  use  of  the 
sensitive  differential  gage  is  desirable. 

In  the  series  of  experiments  referred  to,  the  results  worked 
out  for  the  resistance  of  fittings  were  more  erratic  than  those  for 
straight  pipes.  Hence  no  claim  is  made  for  precision  or  finality 
in  the  results  here  presented.  However,  two  important  conclu- 
sions are  reached.  One  is  that  the  resistance  of  globe  valves  has 
heretofore  been  underestimated,  and  the  importance  of  reaming 
small  pipe  has  not  been  appreciated. 

TABLE  OF  LENGTHS  OF  PIPE  IN  FEET  THAT  GIVE  RESISTANCE  EQUAL  THAT 
OF  VARIOUS  FITTINGS 


Diameter 
of  Pipe 

90°  Elbows 

Unreamed 
Joints,  Two 
Ends 

Reamed  Joints, 
Two  Ends 

Return 
Bends 

Globe 
Valves 

| 

10.0 

2  to  4 

1.0 

10.0 

20.0 

1 

7.0 

ii 

1.0 

7.0 

25.0 

1 

5.0 

i  i 

1.0 

5.0 

40.0 

1| 

4.0 

ct 

1.0 

4.0 

45.0 

2 

3.5 

« 

1.0 

3.5 

47.0 

A  series  of  runs  was  made  on  50-foot  lengths  of  rubber-lined 
armored  hose  such  as  is  used  to  connect  with  compressed-air  tools. 
The  scheme  was  the  same  as  that  described  for  pipes  and  fittings ; 

v  2 
and  the  range  of  —  was  the  same.     The  average  results  are  here 

given.  This  includes  the  resistance  in  a  50-foot  length  with  the 
metallic  end  couplings.  In  these  end  connections  a  considerable 
contraction  occurs.  For  the  half-inch  hose  the  end  couplings  are 
quarter-inch.  The  excessive  resistance  in  the  half-inch  hose  may 
have  been  due  to  these  end  contractions  or  to  some  other  ob- 
struction. It  is  a  further  illustration  of  the  fact  that  reliable 
coefficients  cannot  be  gotten  for  pipes  of  half -inch  diameter  and 
less. 


Diameter  of  hose  in  inches 
Resistance  in  50-ft.  lengths 

i 

950.0^ 
r 

a 

4 

20.0^ 
r 

1 

4.5^ 
r 

H 

2-6  7° 

APPENDIX  D 
THE  OIL  DIFFERENTIAL  GAGE 

Examination  of  Eq.  (21)  shows  that  the  greatest  liability  to 
inaccuracy  lies  in  determining  i  since  it  is  relatively  small  as  com- 
pared with  t  and  p  and  the  conditions  are  such  that  the  scale  can- 
not be  read  with  precision.  To  better  determine  i  the  oil  differ- 
ential gage  may  be  used.  A  special  design  suitable  to  this 
purpose  is  illustrated  in  Fig.  32. 1 

The  special  fittings  are  inserted  in  place  of  the  two  plain  glass 
tubes  of  Fig.  32.  The  cocks  being  lettered  similarly  in  the  two 
figures. 

The  special  features  of  the  gage  are  the  two  reservoirs  G\  and 
<72  the  capacity  of  each  being  controlled  by  the  movable  piston. 

The  manipulation  is  as  follows :  With  water  in  the  low  pressure 
side  and  oil  in  the  high  pressure  side  and  with  C2  and  C3  open,  the 

specific  gravity  of  the  oil  is  -^-  =  s.     Now  with  Cz  and  C$  closed 

^o 

and  C4  and  C$  open  the  higher  pressure  will  depress  the  surface, 
Ai,  of  the  oil  and  raise  A2,  that  of  the  water.  Now,  by  manipu- 
lating the  piston  Gi  oil  can  be  forced  in  or  withdrawn  from  the 
gage  tube  until  AL  and  A\  are  in  the  same  horizontal  plane,  or  on 
the  same  scale  line.  While  this  coincidence  holds  i  =  Z0  (1  —  s). 
Proof. — Let  w  equal  pressure  due  to  1  in.  of  water  head  and 
0  equal  that  due  to  1  in.  of  oil  head.  Then  since  the  two 
pressures  become  equal  on  the  line  BB,  we  have 

p  -f-  OZ0  =  pz  +  wZ0  and  p\  —  p%  =  ZQ  (w  —  0) 

but  i  =  Pl  ~  P-     Therefore,  i  =  Z  (1  -  s). 

With  kerosene  oil  in  the  gage  i  equals  one-fifth  of  ZQ  very  nearly. 

The  length  of  oil  and  water  in  the  gage  tubes  can  be  further  con- 
trolled by  the  drain  cocks  on  the  reservoirs.  The  length  of  oil 
should  be  about  five  times  as  much  as  the  anticipated  i  and  this 
(i)  must  be  kept  within  the  limits  specified  by  the  standards  of 

1  This  gage  was  designed  and  is  used  in  the  laboratories  of  the  Missouri 
School  of  Mines. 

187 


188 


APPENDIX  D 


FIG.  32. 


APPENDIX  D  189 

practice.     Say  within  12  in.     Gage  tubes  about  4  ft.  long  will  be 
found  convenient. 

The  small  pipes  e\  and  ez  connect  with  the  air  main  and  so 
practically  balance  the  pressures  on  the  pistons  in  the  reservoirs. 
Thus  their  movements  are  made  easier  and  leakage  by  the  pistons 
practically  eliminated. 


INDEX 


Adiabatic  compression,  3 
Adjustment  of  valves,  15 
Air-lift  pump — Chart,  81 

Data  on,  85 

Designing,  78 

Dredge,  83 

Testing  wells,  84 

Theory,  76 

Air  used    without   expansion — Effi- 
ciency, 30 
Altitude — Variation  of  pressure  with, 

31 
Atmospheric  air — Weights,  131 

Pressure — Variation   with  alti- 
tude, 31 

B 

Blowers — Rotary,  113 
C 

Centrifugal  air  compressors,  105 

Chart — Air-lift  pump,  81 

Design  of  logarithmic,  172 
Friction  in  air  pipes,  53 
Friction  coefficients,  52 
Performance  of  fans,  102 

Clearance — Effect  in  Compression,  13 
Expansion  engines,  14 
From  indicator  cards,  11 

Coefficients  for  friction  formulas,  52 
Orifices,  38,  139 

Compounding,  23 

Proportions  for,  25 

Compounding — Work    in    compres- 
sion, 27,  136 

Cooling — Effect  on  efficiency,  21 

Cooling  water  required,  20 

Compressed  air  reservoirs,  89 


Density  of  air,  19,  131 
Displacement  pumps,  75 
Drill  capacities,  169 


Efficiency-Power  transmission,  60 

Volumetric,  13,  17 

Working  without  expansion,  30, 

138 

Elevation  and  pressure  table,  138 
Exhaust  pumps,  29 


Fans,  91 

Designing,  98 
Performance  chart,  102 
Suggestions  on  designing,  103 
Flow  through  orifices,  35-44 
Friction — In  air  pipes,  49-55 
Chart  for,  53 
Experiments  in,  179 
Table  for,  141 
Gas  pipes,  55 
In  pipe  fittings,  55 


H 


Heating — Effect  on  Volumetric  effi- 
ciency, 17 

Effect  on  work  efficiency,  21 
Hydraulic     air     compressors — Dis- 
placement type,  63,  139 
Entanglement  type,  64 


Indicator  cards,  9 
Isothermal  compression,  1 


191 


192 


INDEX 


Logarithms — Hyperbolic,  147 
M 

Measurement  of  air,  33 
By  standard  orifice,  35 
In  closed  tanks,  46 
Under  pressure,  41 

Moisture  in  air,  19 
Tables,  134 

O 

Oil  differential  gage,  187 
Orifice — Apparatus,  37 
Chart  for  sizes,  39 
Coefficients,  38,  43 
Tables  of,  139,  140 


Pipes — Dimensions  of,  146 

Planimeter  constants,  9 

Power  transmission  by  compressed 

air,  60 

Pump— Air-lift,  76 
Pump — Design  of  a  mine  pump,  115 

Design  of  displacement  pump, 
120 

Displacement,  75 

Exhaust,  29 


Reheating,  21 


Return-air  pump,  69 
Return-air  system,  68 

S 

Standard  orifice,  35 
T 

Tables — See  Subjects. 
Temperature  changes,  18 
Temperature  Table,  129 
Turbo    air   compressors — Formulas, 

107 
Proportions,  111 


Valve — Adjustments,  15 
Variable  intake  pressure,  27 
Volumetric  efficiency,  13,  17 
Volume — Pressure,    Temperature, 
Work,  Table,  129 

W 

Water-cooling,  20 

Water  gages,  40 

Weight  of  air,  19,  131 

Work — Adiabatic,  3 

In  terms  of  temperature,  4 
Incomplete  expansion,  7 
Isothermal  conditions,  1 
Variable  intake  pressure,  27 


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